Results of Orthogonal Polynomials I

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.1#Ex7	${\displaystyle{\displaystyle\left(z;q\right)_{0}=1}}$ \qPochhammer{z}{q}{0} = 1		QPochhammer(z, q, 0) = 1	QPochhammer[z, q, 0] == 1	Successful	Successful	-	Successful [Tested: 70]
18.1#Ex10	${\displaystyle{\displaystyle\left(z;q\right)_{\infty}=\prod_{j=0}^{\infty}(1-zq^{j})}}$ \qPochhammer{z}{q}{\infty} = \prod_{j=0}^{\infty}(1-zq^{j})		QPochhammer(z, q, infinity) = product(1 - z*(q)^(j), j = 0..infinity)	QPochhammer[z, q, Infinity] == Product[1 - z*(q)^(j), {j, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [56 / 70] Result: Plus[Times[-1.0, QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]]], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Times[-1.0, QPochhammer[Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]]], QPochhammer[Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
18.1.E1	${\displaystyle{\displaystyle C^{(0)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)}}$ \ultrasphpoly{0}{n}@{x} = \frac{2}{n}\ChebyshevpolyT{n}@{x}		GegenbauerC(n, 0, x) = (2)/(n)*ChebyshevT(n, x)	GegenbauerC[n, 0, x] == Divide[2,n]*ChebyshevT[n, x]	Failure	Failure	Successful [Tested: 3]	Failed [3 / 3] Result: -6.0 Test Values: {Rule[n, 3], Rule[x, 1.5]} Result: 0.6666666666666666 Test Values: {Rule[n, 3], Rule[x, 0.5]} ... skip entries to safe data
18.1.E1	${\displaystyle{\displaystyle\frac{2}{n}T_{n}\left(x\right)=\frac{2(n-1)!}{{\left(\tfrac{1}{2}\right)_{n}}}P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}}$ \frac{2}{n}\ChebyshevpolyT{n}@{x} = \frac{2(n-1)!}{\Pochhammersym{\tfrac{1}{2}}{n}}\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}		(2)/(n)*ChebyshevT(n, x) = (2*factorial(n - 1))/(pochhammer((1)/(2), n))*JacobiP(n, -(1)/(2), -(1)/(2), x)	Divide[2,n]*ChebyshevT[n, x] == Divide[2*(n - 1)!,Pochhammer[Divide[1,2], n]]*JacobiP[n, -Divide[1,2], -Divide[1,2], x]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 3]
18.1.E2	${\displaystyle{\displaystyle G_{n}\left(p,q,x\right)=\frac{n!}{{\left(n+p\right)_{n}}}P^{(p-q,q-1)}_{n}\left(2x-1\right)}}$ \shiftJacobipolyG{n}@{p}{q}{x} = \frac{n!}{\Pochhammersym{n+p}{n}}\JacobipolyP{p-q}{q-1}{n}@{2x-1}		JacobiP(n, p-q, q-1, 2*(x)-1)*((n)!)/pochhammer(n+p, n) = (factorial(n))/(pochhammer(n + p, n))*JacobiP(n, p - q, q - 1, 2*x - 1)	Error	Successful	Missing Macro Error	-	-
18.2.E1	${\displaystyle{\displaystyle\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x=0}}$ \int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\diff{x} = 0	${\displaystyle{\displaystyle n\neq m}}$	int(p[n](x)* p[m](x)* w(x), x = a..b) = 0	Integrate[Subscript[p, n][x]* Subscript[p, m][x]* w[x], {x, a, b}, GenerateConditions->None] == 0	Failure	Failure	Successful [Tested: 300]	Successful [Tested: 300]
18.2.E2	${\displaystyle{\displaystyle\sum_{x\in X}p_{n}(x)p_{m}(x)w_{x}=0}}$ \sum_{x\in X}p_{n}(x)p_{m}(x)w_{x} = 0	${\displaystyle{\displaystyle n\neq m}}$	sum(p[n](x)* p[m](x)* w[x], x in X) = 0	Sum[Subscript[p, n][x]* Subscript[p, m][x]* Subscript[w, x], {x, X}, GenerateConditions->None] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
18.2.E3	${\displaystyle{\displaystyle\sum_{x\in X}p_{n}(x)p_{m}(x)w_{x}=0}}$ \sum_{x\in X}p_{n}(x)p_{m}(x)w_{x} = 0		sum(p[n](x)* p[m](x)* w[x], x in X) = 0	Sum[Subscript[p, n][x]* Subscript[p, m][x]* Subscript[w, x], {x, X}, GenerateConditions->None] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
18.2.E4	${\displaystyle{\displaystyle\sum_{x\in X}x^{2n}w_{x}<\infty}}$ \sum_{x\in X}x^{2n}w_{x} < \infty		sum((x)^(2*n)* w[x](<)*infinity, x in X)	Sum[(x)^(2*n)* Subscript[w, x][<]*Infinity, {x, X}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.2.E8	${\displaystyle{\displaystyle p_{n+1}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{n-1}(x)}}$ p_{n+1}(x) = (A_{n}x+B_{n})p_{n}(x)-C_{n}p_{n-1}(x)	${\displaystyle{\displaystyle n\geq 0}}$	p[n + 1](x) = (((k[n + 1])/(k[n]))*x + B[n])*p[n](x)- C[n]*p[n - 1](x)	Subscript[p, n + 1][x] == ((Divide[Subscript[k, n + 1],Subscript[k, n]])*x + Subscript[B, n])*Subscript[p, n][x]- Subscript[C, n]*Subscript[p, n - 1][x]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.3.E1	${\displaystyle{\displaystyle\sum_{n=1}^{N+1}T_{j}\left(x_{N+1,n}\right)T_{k}\left(x_{N+1,n}\right)=0}}$ \sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0	${\displaystyle{\displaystyle 0\leq j,j\leq N,0\leq k,k\leq N,j\neq k}}$	sum(ChebyshevT(j, x[N + 1 , n])*ChebyshevT(k, x[N + 1 , n]), n = 1..N + 1) = 0	Sum[ChebyshevT[j, Subscript[x, N + 1 , n]]*ChebyshevT[k, Subscript[x, N + 1 , n]], {n, 1, N + 1}, GenerateConditions->None] == 0	Skipped - Unable to analyze test case: Null	Skipped - Unable to analyze test case: Null	-	-
18.3.E2	${\displaystyle{\displaystyle x_{N+1,n}=\cos\left((n-\tfrac{1}{2})\pi/(N+1)\right)}}$ x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}		x[N + 1 , n] = cos((n -(1)/(2))*Pi/(N + 1))	Subscript[x, N + 1 , n] == Cos[(n -Divide[1,2])*Pi/(N + 1)]	Failure	Failure	Failed [298 / 300] Result: .1432026267+.3500908026*I Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 1} Result: 1.718798807+.233214116e-1*I Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [298 / 300] Result: Complex[0.14320262643759762, 0.350090802645732] Test Values: {Rule[n, 1], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, N], n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.7187988066024098, 0.023321411689447014] Test Values: {Rule[n, 2], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, N], n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.5.E1	${\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n\theta\right)}}$ \ChebyshevpolyT{n}@{x} = \cos@{n\theta}		ChebyshevT(n, x) = cos(n*theta)	ChebyshevT[n, x] == Cos[n*\[Theta]]	Failure	Failure	Failed [90 / 90] Result: .7694569811+.3969495503*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1} Result: 3.747751686+1.159954891*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[0.7694569809427748, 0.3969495502290325] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.747751685467572, 1.1599548913509004] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.5.E2	${\displaystyle{\displaystyle U_{n}\left(x\right)=\ifrac{(\sin(n+1)\theta)}{\sin\theta}}}$ \ChebyshevpolyU{n}@{x} = \ifrac{(\sin@@{(n+1)\theta})}{\sin@@{\theta}}		ChebyshevU(n, x) = (sin((n + 1)*theta))/(sin(theta))	ChebyshevU[n, x] == Divide[Sin[(n + 1)*\[Theta]],Sin[\[Theta]]]	Failure	Failure	Failed [90 / 90] Result: 1.538913962+.7938991006*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1} Result: 7.495503373+2.319909783*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[1.5389139618855496, 0.7938991004580651] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[7.495503370935143, 2.3199097827018003] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.5.E6	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\frac{1}{x}\right)=\frac{(-1)^{n}}{n!}x^{n+\alpha+1}e^{\ifrac{1}{x}}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{-\alpha-1}e^{-\ifrac{1}{x}}\right)}}$ \LaguerrepolyL[\alpha]{n}@{\frac{1}{x}} = \frac{(-1)^{n}}{n!}x^{n+\alpha+1}e^{\ifrac{1}{x}}\deriv[n]{}{x}\left(x^{-\alpha-1}e^{-\ifrac{1}{x}}\right)		LaguerreL(n, alpha, (1)/(x)) = ((- 1)^(n))/(factorial(n))*(x)^(n + alpha + 1)* exp((1)/(x))*diff((x)^(- alpha - 1)* exp(-(1)/(x)), [x$(n)])	LaguerreL[n, \[Alpha], Divide[1,x]] == Divide[(- 1)^(n),(n)!]*(x)^(n + \[Alpha]+ 1)* Exp[Divide[1,x]]*D[(x)^(- \[Alpha]- 1)* Exp[-Divide[1,x]], {x, n}]	Missing Macro Error	Failure	-	Failed [24 / 27] Result: Plus[1.8333333333333335, Times[1.9477340410546757, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], 1.5, []], Times[Plus[-1, Times[-1, ], 1], Plus[, Times[-1, 1], Times[-2, , 1.5], Times[-3, Power[, 2], 1.5], Times[2, 1, 1.5], Times[3, , 1, 1.5], Times[-1, 1.5], Times[2, 1.5, 1.5], Times[2, , 1.5, 1.5]], [Plus[1, ]]], Times[-1, Plus[Times[-1, ], 1, 1.5], Plus[1, , Times[-1, 1], Times[-4, 1.5], Times[-7, , 1.5], Times[-3, Power[, 2], 1.5], Times[4, 1, 1.5], Times[3, , 1, 1.5], Times[2, 1.5, 1.5], Times[, 1.5, 1.5]], [Plus[2, ]]], Times[Plus[2, ], 1.5, Plus[-1, Times[-1, ], 1, 1.5], Plus[Times[-1, ], 1, 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], Times[Power[E, Times[-1, Power[1.5, -1]]], Binomial[Plus[-1, Times[-1, 1.5]], 1]]]}]][2.0]]], {Rule[n, 1], Rule[x, 1.5], Rule[α, 1.5]} Result: Plus[2.2638888888888893, Times[-1.9477340410546757, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], 1.5, []], Times[Plus[-1, Times[-1, ], 2], Plus[, Times[-1, 2], Times[-2, , 1.5], Times[-3, Power[, 2], 1.5], Times[2, 2, 1.5], Times[3, , 2, 1.5], Times[-1, 1.5], Times[2, 1.5, 1.5], Times[2, , 1.5, 1.5]], [Plus[1, ]]], Times[-1, Plus[Times[-1, ], 2, 1.5], Plus[1, , Times[-1, 2], Times[-4, 1.5], Times[-7, , 1.5], Times[-3, Power[, 2], 1.5], Times[4, 2, 1.5], Times[3, , 2, 1.5], Times[2, 1.5, 1.5], Times[, 1.5, 1.5]], [Plus[2, ]]], Times[Plus[2, ], 1.5, Plus[-1, Times[-1, ], 2, 1.5], Plus[Times[-1, ], 2, 1.5], [Plus[3, ]]]], 0], Equal[[-1], 0], Equal[[0], 0], Equal[[1], Times[Power[E, Times[-1, Power[1.5, -1]]], Binomial[Plus[-1, Times[-1, 1.5]], 2]]]}]][3.0]]], {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5]} ... skip entries to safe data
18.5.E7	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}}}$ \JacobipolyP{\alpha}{\beta}{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}		JacobiP(n, alpha, beta, x) = sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n)	JacobiP[n, \[Alpha], \[Beta], x] == Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 81]
18.5.E7	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+\beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right)}}$ \sum_{\ell=0}^{n}\frac{\Pochhammersym{n+\alpha+\beta+1}{\ell}\Pochhammersym{\alpha+\ell+1}{n-\ell}}{\ell!\;(n-\ell)!}\left(\frac{x-1}{2}\right)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{2}{1}@@{-n,n+\alpha+\beta+1}{\alpha+1}{\frac{1-x}{2}}		sum((pochhammer(n + alpha + beta + 1, ell)*pochhammer(alpha + ell + 1, n - ell))/(factorial(ell)*factorial(n - ell))*((x - 1)/(2))^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n , n + alpha + beta + 1], [alpha + 1], (1 - x)/(2))	Sum[Divide[Pochhammer[n + \[Alpha]+ \[Beta]+ 1, \[ScriptL]]*Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(Divide[x - 1,2])^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n , n + \[Alpha]+ \[Beta]+ 1}, {\[Alpha]+ 1}, Divide[1 - x,2]]	Successful	Successful	-	Successful [Tested: 81]
18.5.E8	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}}}$ \JacobipolyP{\alpha}{\beta}{n}@{x} = 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}		JacobiP(n, alpha, beta, x) = (2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n)	JacobiP[n, \[Alpha], \[Beta], x] == (2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 81]	Successful [Tested: 81]
18.5.E8	${\displaystyle{\displaystyle 2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}{{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right)}}$ 2^{-n}\sum_{\ell=0}^{n}\binom{n+\alpha}{\ell}\binom{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\left(\frac{x+1}{2}\right)^{n}\genhyperF{2}{1}@@{-n,-n-\beta}{\alpha+1}{\frac{x-1}{x+1}}		(2)^(- n)* sum(binomial(n + alpha,ell)*binomial(n + beta,n - ell)*(x - 1)^(n - ell)*(x + 1)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*((x + 1)/(2))^(n)* hypergeom([- n , - n - beta], [alpha + 1], (x - 1)/(x + 1))	(2)^(- n)* Sum[Binomial[n + \[Alpha],\[ScriptL]]*Binomial[n + \[Beta],n - \[ScriptL]]*(x - 1)^(n - \[ScriptL])*(x + 1)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(Divide[x + 1,2])^(n)* HypergeometricPFQ[{- n , - n - \[Beta]}, {\[Alpha]+ 1}, Divide[x - 1,x + 1]]	Failure	Failure	Successful [Tested: 81]	Successful [Tested: 81]
18.5.E9	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,n+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-x}{2}\right)}}$ \ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,n+2\lambda}{\lambda+\tfrac{1}{2}}{\frac{1-x}{2}}		GegenbauerC(n, lambda, x) = (pochhammer(2*lambda, n))/(factorial(n))*hypergeom([- n , n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2))	GegenbauerC[n, \[Lambda], x] == Divide[Pochhammer[2*\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , n + 2*\[Lambda]}, {\[Lambda]+Divide[1,2]}, Divide[1 - x,2]]	Successful	Successful	-	Failed [15 / 90] Result: 0.375 Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, -1.5]} Result: 0.4375 Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -1.5]} ... skip entries to safe data
18.5.E10	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}}}$ \ultrasphpoly{\lambda}{n}@{x} = \sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}		GegenbauerC(n, lambda, x) = sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/2))	GegenbauerC[n, \[Lambda], x] == Sum[Divide[(- 1)^\[ScriptL]* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]	Failure	Successful	Manual Skip!	Successful [Tested: 90]
18.5.E10	${\displaystyle{\displaystyle\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right)}}$ \sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell} = (2x)^{n}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{1-\lambda-n}{\frac{1}{x^{2}}}		sum(((- 1)^(ell)* pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - 2*ell))*(2*x)^(n - 2*ell), ell = 0..floor(n/2)) = (2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [1 - lambda - n], (1)/((x)^(2)))	Sum[Divide[(- 1)^\[ScriptL]* Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - 2*\[ScriptL])!]*(2*x)^(n - 2*\[ScriptL]), {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None] == (2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {1 - \[Lambda]- n}, Divide[1,(x)^(2)]]	Failure	Failure	Manual Skip!	Failed [3 / 90] Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 0.5], Rule[λ, -2]} ... skip entries to safe data
18.5.E11	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)}}$ \ultrasphpoly{\lambda}{n}@{\cos@@{\theta}} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta}		GegenbauerC(n, lambda, cos(theta)) = sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)*theta), ell = 0..n)	GegenbauerC[n, \[Lambda], Cos[\[Theta]]] == Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None]	Failure	Failure	Error	Failed [30 / 300] Result: Indeterminate Test Values: {Rule[n, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} ... skip entries to safe data
18.5.E11	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)=e^{\mathrm{i}n\theta}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};e^{-2\mathrm{i}\theta}\right)}}$ \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\;(n-\ell)!}\cos@{(n-2\ell)\theta} = e^{\iunit n\theta}\frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,\lambda}{1-\lambda-n}{e^{-2\iunit\theta}}		sum((pochhammer(lambda, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*cos((n - 2*ell)*theta), ell = 0..n) = exp(I*n*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda], [1 - lambda - n], exp(- 2*I*theta))	Sum[Divide[Pochhammer[\[Lambda], \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*Cos[(n - 2*\[ScriptL])*\[Theta]], {\[ScriptL], 0, n}, GenerateConditions->None] == Exp[I*n*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]}, {1 - \[Lambda]- n}, Exp[- 2*I*\[Theta]]]	Failure	Failure	Error	Failed [30 / 300] Result: Indeterminate Test Values: {Rule[n, 1], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} ... skip entries to safe data
18.5.E12	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1\right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}}}$ \LaguerrepolyL[\alpha]{n}@{x} = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\alpha+\ell+1}{n-\ell}}{(n-\ell)!\;\ell!}(-x)^{\ell}		LaguerreL(n, alpha, x) = sum((pochhammer(alpha + ell + 1, n - ell))/(factorial(n - ell)*factorial(ell))*(- x)^(ell), ell = 0..n)	LaguerreL[n, \[Alpha], x] == Sum[Divide[Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(n - \[ScriptL])!*(\[ScriptL])!]*(- x)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 27]
18.5.E12	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1\right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{1}F_{1}}\left({-n\atop\alpha+1};x\right)}}$ \sum_{\ell=0}^{n}\frac{\Pochhammersym{\alpha+\ell+1}{n-\ell}}{(n-\ell)!\;\ell!}(-x)^{\ell} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\genhyperF{1}{1}@@{-n}{\alpha+1}{x}		sum((pochhammer(alpha + ell + 1, n - ell))/(factorial(n - ell)*factorial(ell))*(- x)^(ell), ell = 0..n) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n], [alpha + 1], x)	Sum[Divide[Pochhammer[\[Alpha]+ \[ScriptL]+ 1, n - \[ScriptL]],(n - \[ScriptL])!*(\[ScriptL])!]*(- x)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n}, {\[Alpha]+ 1}, x]	Successful	Successful	-	Successful [Tested: 27]
18.5.E13	${\displaystyle{\displaystyle H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}}}$ \HermitepolyH{n}@{x} = n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}		HermiteH(n, x) = factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/2))	HermiteH[n, x] == (n)!*Sum[Divide[(- 1)^\[ScriptL]*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.5.E13	${\displaystyle{\displaystyle n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right)}}$ n!\sum_{\ell=0}^{\floor{n/2}}\frac{(-1)^{\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!} = (2x)^{n}\genhyperF{2}{0}@@{-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}}{-}{-\frac{1}{x^{2}}}		factorial(n)*sum(((- 1)^(ell)*(2*x)^(n - 2*ell))/(factorial(ell)*factorial(n - 2*ell)), ell = 0..floor(n/2)) = (2*x)^(n)* hypergeom([-(1)/(2)*n , -(1)/(2)*n +(1)/(2)], [-], -(1)/((x)^(2)))	(n)!*Sum[Divide[(- 1)^\[ScriptL]*(2*x)^(n - 2*\[ScriptL]),(\[ScriptL])!*(n - 2*\[ScriptL])!], {\[ScriptL], 0, Floor[n/2]}, GenerateConditions->None] == (2*x)^(n)* HypergeometricPFQ[{-Divide[1,2]*n , -Divide[1,2]*n +Divide[1,2]}, {-}, -Divide[1,(x)^(2)]]	Error	Failure	Skip - symbolical successful subtest	Error
18.5#Ex1	${\displaystyle{\displaystyle T_{0}\left(x\right)=1}}$ \ChebyshevpolyT{0}@{x} = 1		ChebyshevT(0, x) = 1	ChebyshevT[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex2	${\displaystyle{\displaystyle T_{1}\left(x\right)=x}}$ \ChebyshevpolyT{1}@{x} = x		ChebyshevT(1, x) = x	ChebyshevT[1, x] == x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex3	${\displaystyle{\displaystyle T_{2}\left(x\right)=2x^{2}-1}}$ \ChebyshevpolyT{2}@{x} = 2x^{2}-1		ChebyshevT(2, x) = 2*(x)^(2)- 1	ChebyshevT[2, x] == 2*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex4	${\displaystyle{\displaystyle T_{3}\left(x\right)=4x^{3}-3x}}$ \ChebyshevpolyT{3}@{x} = 4x^{3}-3x		ChebyshevT(3, x) = 4*(x)^(3)- 3*x	ChebyshevT[3, x] == 4*(x)^(3)- 3*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex5	${\displaystyle{\displaystyle T_{4}\left(x\right)=8x^{4}-8x^{2}+1}}$ \ChebyshevpolyT{4}@{x} = 8x^{4}-8x^{2}+1		ChebyshevT(4, x) = 8*(x)^(4)- 8*(x)^(2)+ 1	ChebyshevT[4, x] == 8*(x)^(4)- 8*(x)^(2)+ 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex6	${\displaystyle{\displaystyle T_{5}\left(x\right)=16x^{5}-20x^{3}+5x}}$ \ChebyshevpolyT{5}@{x} = 16x^{5}-20x^{3}+5x		ChebyshevT(5, x) = 16*(x)^(5)- 20*(x)^(3)+ 5*x	ChebyshevT[5, x] == 16*(x)^(5)- 20*(x)^(3)+ 5*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex7	${\displaystyle{\displaystyle T_{6}\left(x\right)=32x^{6}-48x^{4}+18x^{2}-1}}$ \ChebyshevpolyT{6}@{x} = 32x^{6}-48x^{4}+18x^{2}-1		ChebyshevT(6, x) = 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1	ChebyshevT[6, x] == 32*(x)^(6)- 48*(x)^(4)+ 18*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex8	${\displaystyle{\displaystyle U_{0}\left(x\right)=1}}$ \ChebyshevpolyU{0}@{x} = 1		ChebyshevU(0, x) = 1	ChebyshevU[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex9	${\displaystyle{\displaystyle U_{1}\left(x\right)=2x}}$ \ChebyshevpolyU{1}@{x} = 2x		ChebyshevU(1, x) = 2*x	ChebyshevU[1, x] == 2*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex10	${\displaystyle{\displaystyle U_{2}\left(x\right)=4x^{2}-1}}$ \ChebyshevpolyU{2}@{x} = 4x^{2}-1		ChebyshevU(2, x) = 4*(x)^(2)- 1	ChebyshevU[2, x] == 4*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex11	${\displaystyle{\displaystyle U_{3}\left(x\right)=8x^{3}-4x}}$ \ChebyshevpolyU{3}@{x} = 8x^{3}-4x		ChebyshevU(3, x) = 8*(x)^(3)- 4*x	ChebyshevU[3, x] == 8*(x)^(3)- 4*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex12	${\displaystyle{\displaystyle U_{4}\left(x\right)=16x^{4}-12x^{2}+1}}$ \ChebyshevpolyU{4}@{x} = 16x^{4}-12x^{2}+1		ChebyshevU(4, x) = 16*(x)^(4)- 12*(x)^(2)+ 1	ChebyshevU[4, x] == 16*(x)^(4)- 12*(x)^(2)+ 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex13	${\displaystyle{\displaystyle U_{5}\left(x\right)=32x^{5}-32x^{3}+6x}}$ \ChebyshevpolyU{5}@{x} = 32x^{5}-32x^{3}+6x		ChebyshevU(5, x) = 32*(x)^(5)- 32*(x)^(3)+ 6*x	ChebyshevU[5, x] == 32*(x)^(5)- 32*(x)^(3)+ 6*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex14	${\displaystyle{\displaystyle U_{6}\left(x\right)=64x^{6}-80x^{4}+24x^{2}-1}}$ \ChebyshevpolyU{6}@{x} = 64x^{6}-80x^{4}+24x^{2}-1		ChebyshevU(6, x) = 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1	ChebyshevU[6, x] == 64*(x)^(6)- 80*(x)^(4)+ 24*(x)^(2)- 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex15	${\displaystyle{\displaystyle P_{0}\left(x\right)=1}}$ \LegendrepolyP{0}@{x} = 1		LegendreP(0, x) = 1	LegendreP[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex16	${\displaystyle{\displaystyle P_{1}\left(x\right)=x}}$ \LegendrepolyP{1}@{x} = x		LegendreP(1, x) = x	LegendreP[1, x] == x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex17	${\displaystyle{\displaystyle P_{2}\left(x\right)=\tfrac{3}{2}x^{2}-\tfrac{1}{2}}}$ \LegendrepolyP{2}@{x} = \tfrac{3}{2}x^{2}-\tfrac{1}{2}		LegendreP(2, x) = (3)/(2)*(x)^(2)-(1)/(2)	LegendreP[2, x] == Divide[3,2]*(x)^(2)-Divide[1,2]	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex18	${\displaystyle{\displaystyle P_{3}\left(x\right)=\tfrac{5}{2}x^{3}-\tfrac{3}{2}x}}$ \LegendrepolyP{3}@{x} = \tfrac{5}{2}x^{3}-\tfrac{3}{2}x		LegendreP(3, x) = (5)/(2)*(x)^(3)-(3)/(2)*x	LegendreP[3, x] == Divide[5,2]*(x)^(3)-Divide[3,2]*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex19	${\displaystyle{\displaystyle P_{4}\left(x\right)=\tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8}}}$ \LegendrepolyP{4}@{x} = \tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8}		LegendreP(4, x) = (35)/(8)*(x)^(4)-(15)/(4)*(x)^(2)+(3)/(8)	LegendreP[4, x] == Divide[35,8]*(x)^(4)-Divide[15,4]*(x)^(2)+Divide[3,8]	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex20	${\displaystyle{\displaystyle P_{5}\left(x\right)=\tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x}}$ \LegendrepolyP{5}@{x} = \tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x		LegendreP(5, x) = (63)/(8)*(x)^(5)-(35)/(4)*(x)^(3)+(15)/(8)*x	LegendreP[5, x] == Divide[63,8]*(x)^(5)-Divide[35,4]*(x)^(3)+Divide[15,8]*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex21	${\displaystyle{\displaystyle P_{6}\left(x\right)=\tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}}}$ \LegendrepolyP{6}@{x} = \tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-\tfrac{5}{16}		LegendreP(6, x) = (231)/(16)*(x)^(6)-(315)/(16)*(x)^(4)+(105)/(16)*(x)^(2)-(5)/(16)	LegendreP[6, x] == Divide[231,16]*(x)^(6)-Divide[315,16]*(x)^(4)+Divide[105,16]*(x)^(2)-Divide[5,16]	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex22	${\displaystyle{\displaystyle L_{0}\left(x\right)=1}}$ \LaguerrepolyL[]{0}@{x} = 1		LaguerreL(0, x) = 1	LaguerreL[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex23	${\displaystyle{\displaystyle L_{1}\left(x\right)=-x+1}}$ \LaguerrepolyL[]{1}@{x} = -x+1		LaguerreL(1, x) = - x + 1	LaguerreL[1, x] == - x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex24	${\displaystyle{\displaystyle L_{2}\left(x\right)=\tfrac{1}{2}x^{2}-2x+1}}$ \LaguerrepolyL[]{2}@{x} = \tfrac{1}{2}x^{2}-2x+1		LaguerreL(2, x) = (1)/(2)*(x)^(2)- 2*x + 1	LaguerreL[2, x] == Divide[1,2]*(x)^(2)- 2*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex25	${\displaystyle{\displaystyle L_{3}\left(x\right)=-\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1}}$ \LaguerrepolyL[]{3}@{x} = -\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1		LaguerreL(3, x) = -(1)/(6)*(x)^(3)+(3)/(2)*(x)^(2)- 3*x + 1	LaguerreL[3, x] == -Divide[1,6]*(x)^(3)+Divide[3,2]*(x)^(2)- 3*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex26	${\displaystyle{\displaystyle L_{4}\left(x\right)=\tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1}}$ \LaguerrepolyL[]{4}@{x} = \tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1		LaguerreL(4, x) = (1)/(24)*(x)^(4)-(2)/(3)*(x)^(3)+ 3*(x)^(2)- 4*x + 1	LaguerreL[4, x] == Divide[1,24]*(x)^(4)-Divide[2,3]*(x)^(3)+ 3*(x)^(2)- 4*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex27	${\displaystyle{\displaystyle L_{5}\left(x\right)=-\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1}}$ \LaguerrepolyL[]{5}@{x} = -\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}-5x+1		LaguerreL(5, x) = -(1)/(120)*(x)^(5)+(5)/(24)*(x)^(4)-(5)/(3)*(x)^(3)+ 5*(x)^(2)- 5*x + 1	LaguerreL[5, x] == -Divide[1,120]*(x)^(5)+Divide[5,24]*(x)^(4)-Divide[5,3]*(x)^(3)+ 5*(x)^(2)- 5*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex28	${\displaystyle{\displaystyle L_{6}\left(x\right)=\tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1}}$ \LaguerrepolyL[]{6}@{x} = \tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1		LaguerreL(6, x) = (1)/(720)*(x)^(6)-(1)/(20)*(x)^(5)+(5)/(8)*(x)^(4)-(10)/(3)*(x)^(3)+(15)/(2)*(x)^(2)- 6*x + 1	LaguerreL[6, x] == Divide[1,720]*(x)^(6)-Divide[1,20]*(x)^(5)+Divide[5,8]*(x)^(4)-Divide[10,3]*(x)^(3)+Divide[15,2]*(x)^(2)- 6*x + 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex29	${\displaystyle{\displaystyle H_{0}\left(x\right)=1}}$ \HermitepolyH{0}@{x} = 1		HermiteH(0, x) = 1	HermiteH[0, x] == 1	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex30	${\displaystyle{\displaystyle H_{1}\left(x\right)=2x}}$ \HermitepolyH{1}@{x} = 2x		HermiteH(1, x) = 2*x	HermiteH[1, x] == 2*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex31	${\displaystyle{\displaystyle H_{2}\left(x\right)=4x^{2}-2}}$ \HermitepolyH{2}@{x} = 4x^{2}-2		HermiteH(2, x) = 4*(x)^(2)- 2	HermiteH[2, x] == 4*(x)^(2)- 2	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex32	${\displaystyle{\displaystyle H_{3}\left(x\right)=8x^{3}-12x}}$ \HermitepolyH{3}@{x} = 8x^{3}-12x		HermiteH(3, x) = 8*(x)^(3)- 12*x	HermiteH[3, x] == 8*(x)^(3)- 12*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex33	${\displaystyle{\displaystyle H_{4}\left(x\right)=16x^{4}-48x^{2}+12}}$ \HermitepolyH{4}@{x} = 16x^{4}-48x^{2}+12		HermiteH(4, x) = 16*(x)^(4)- 48*(x)^(2)+ 12	HermiteH[4, x] == 16*(x)^(4)- 48*(x)^(2)+ 12	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex34	${\displaystyle{\displaystyle H_{5}\left(x\right)=32x^{5}-160x^{3}+120x}}$ \HermitepolyH{5}@{x} = 32x^{5}-160x^{3}+120x		HermiteH(5, x) = 32*(x)^(5)- 160*(x)^(3)+ 120*x	HermiteH[5, x] == 32*(x)^(5)- 160*(x)^(3)+ 120*x	Successful	Successful	-	Successful [Tested: 3]
18.5#Ex35	${\displaystyle{\displaystyle H_{6}\left(x\right)=64x^{6}-480x^{4}+720x^{2}-120}}$ \HermitepolyH{6}@{x} = 64x^{6}-480x^{4}+720x^{2}-120		HermiteH(6, x) = 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120	HermiteH[6, x] == 64*(x)^(6)- 480*(x)^(4)+ 720*(x)^(2)- 120	Successful	Successful	-	Successful [Tested: 3]
18.6.E1	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(0\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}}}$ \LaguerrepolyL[\alpha]{n}@{0} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}		LaguerreL(n, alpha, 0) = (pochhammer(alpha + 1, n))/(factorial(n))	LaguerreL[n, \[Alpha], 0] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]	Missing Macro Error	Successful	-	Successful [Tested: 9]
18.6.E2	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}=\left(\frac{1+x}{2}\right)^{n}}}$ \lim_{\alpha\to\infty}\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{1}} = \left(\frac{1+x}{2}\right)^{n}		limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, 1)), alpha = infinity) = ((1 + x)/(2))^(n)	Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], 1]], \[Alpha] -> Infinity, GenerateConditions->None] == (Divide[1 + x,2])^(n)	Failure	Aborted	Successful [Tested: 27]	Skipped - Because timed out
18.6.E3	${\displaystyle{\displaystyle\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{P^{(\alpha,\beta)}_{n}\left(-1\right)}=\left(\frac{1-x}{2}\right)^{n}}}$ \lim_{\beta\to\infty}\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\JacobipolyP{\alpha}{\beta}{n}@{-1}} = \left(\frac{1-x}{2}\right)^{n}		limit((JacobiP(n, alpha, beta, x))/(JacobiP(n, alpha, beta, - 1)), beta = infinity) = ((1 - x)/(2))^(n)	Limit[Divide[JacobiP[n, \[Alpha], \[Beta], x],JacobiP[n, \[Alpha], \[Beta], - 1]], \[Beta] -> Infinity, GenerateConditions->None] == (Divide[1 - x,2])^(n)	Failure	Failure	Error	Successful [Tested: 27]
18.6.E4	${\displaystyle{\displaystyle\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{(\lambda)}_{n}\left(1\right)}=x^{n}}}$ \lim_{\lambda\to\infty}\frac{\ultrasphpoly{\lambda}{n}@{x}}{\ultrasphpoly{\lambda}{n}@{1}} = x^{n}		limit((GegenbauerC(n, lambda, x))/(GegenbauerC(n, lambda, 1)), lambda = infinity) = (x)^(n)	Limit[Divide[GegenbauerC[n, \[Lambda], x],GegenbauerC[n, \[Lambda], 1]], \[Lambda] -> Infinity, GenerateConditions->None] == (x)^(n)	Failure	Aborted	Successful [Tested: 9]	Skipped - Because timed out
18.6.E5	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\frac{L^{(\alpha)}_{n}\left(\alpha x\right)}{L^{(\alpha)}_{n}\left(0\right)}=(1-x)^{n}}}$ \lim_{\alpha\to\infty}\frac{\LaguerrepolyL[\alpha]{n}@{\alpha x}}{\LaguerrepolyL[\alpha]{n}@{0}} = (1-x)^{n}		limit((LaguerreL(n, alpha, alpha*x))/(LaguerreL(n, alpha, 0)), alpha = infinity) = (1 - x)^(n)	Limit[Divide[LaguerreL[n, \[Alpha], \[Alpha]*x],LaguerreL[n, \[Alpha], 0]], \[Alpha] -> Infinity, GenerateConditions->None] == (1 - x)^(n)	Missing Macro Error	Aborted	-	Skipped - Because timed out
18.7.E1	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)}}$ \ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{\Pochhammersym{\lambda+\frac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{\lambda-\frac{1}{2}}{n}@{x}		GegenbauerC(n, lambda, x) = (pochhammer(2*lambda, n))/(pochhammer(lambda +(1)/(2), n))*JacobiP(n, lambda -(1)/(2), lambda -(1)/(2), x)	GegenbauerC[n, \[Lambda], x] == Divide[Pochhammer[2*\[Lambda], n],Pochhammer[\[Lambda]+Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], \[Lambda]-Divide[1,2], x]	Successful	Successful	-	Failed [15 / 90] Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, -1.5]} Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -1.5]} ... skip entries to safe data
18.7.E2	${\displaystyle{\displaystyle P^{(\alpha,\alpha)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{{\left(2\alpha+1\right)_{n}}}C^{(\alpha+\frac{1}{2})}_{n}\left(x\right)}}$ \JacobipolyP{\alpha}{\alpha}{n}@{x} = \frac{\Pochhammersym{\alpha+1}{n}}{\Pochhammersym{2\alpha+1}{n}}\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{x}		JacobiP(n, alpha, alpha, x) = (pochhammer(alpha + 1, n))/(pochhammer(2*alpha + 1, n))*GegenbauerC(n, alpha +(1)/(2), x)	JacobiP[n, \[Alpha], \[Alpha], x] == Divide[Pochhammer[\[Alpha]+ 1, n],Pochhammer[2*\[Alpha]+ 1, n]]*GegenbauerC[n, \[Alpha]+Divide[1,2], x]	Successful	Successful	-	Successful [Tested: 27]
18.7.E3	${\displaystyle{\displaystyle T_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(1\right)}}}$ \ChebyshevpolyT{n}@{x} = \ifrac{\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}}{\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{1}}		ChebyshevT(n, x) = (JacobiP(n, -(1)/(2), -(1)/(2), x))/(JacobiP(n, -(1)/(2), -(1)/(2), 1))	ChebyshevT[n, x] == Divide[JacobiP[n, -Divide[1,2], -Divide[1,2], x],JacobiP[n, -Divide[1,2], -Divide[1,2], 1]]	Successful	Successful	-	Successful [Tested: 9]
18.7.E4	${\displaystyle{\displaystyle U_{n}\left(x\right)=C^{(1)}_{n}\left(x\right)}}$ \ChebyshevpolyU{n}@{x} = \ultrasphpoly{1}{n}@{x}		ChebyshevU(n, x) = GegenbauerC(n, 1, x)	ChebyshevU[n, x] == GegenbauerC[n, 1, x]	Successful	Successful	-	Successful [Tested: 9]
18.7.E4	${\displaystyle{\displaystyle C^{(1)}_{n}\left(x\right)=\ifrac{(n+1)P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(1\right)}}}$ \ultrasphpoly{1}{n}@{x} = \ifrac{(n+1)\JacobipolyP{\frac{1}{2}}{\frac{1}{2}}{n}@{x}}{\JacobipolyP{\frac{1}{2}}{\frac{1}{2}}{n}@{1}}		GegenbauerC(n, 1, x) = ((n + 1)*JacobiP(n, (1)/(2), (1)/(2), x))/(JacobiP(n, (1)/(2), (1)/(2), 1))	GegenbauerC[n, 1, x] == Divide[(n + 1)*JacobiP[n, Divide[1,2], Divide[1,2], x],JacobiP[n, Divide[1,2], Divide[1,2], 1]]	Successful	Successful	-	Successful [Tested: 9]
18.7.E9	${\displaystyle{\displaystyle P_{n}\left(x\right)=C^{(\frac{1}{2})}_{n}\left(x\right)}}$ \LegendrepolyP{n}@{x} = \ultrasphpoly{\frac{1}{2}}{n}@{x}		LegendreP(n, x) = GegenbauerC(n, (1)/(2), x)	LegendreP[n, x] == GegenbauerC[n, Divide[1,2], x]	Successful	Successful	-	Successful [Tested: 9]
18.7.E9	${\displaystyle{\displaystyle C^{(\frac{1}{2})}_{n}\left(x\right)=P^{(0,0)}_{n}\left(x\right)}}$ \ultrasphpoly{\frac{1}{2}}{n}@{x} = \JacobipolyP{0}{0}{n}@{x}		GegenbauerC(n, (1)/(2), x) = JacobiP(n, 0, 0, x)	GegenbauerC[n, Divide[1,2], x] == JacobiP[n, 0, 0, x]	Successful	Successful	-	Successful [Tested: 9]
18.7.E10	${\displaystyle{\displaystyle P^{*}_{n}\left(x\right)=P_{n}\left(2x-1\right)}}$ \shiftLegendrepolyP{n}@{x} = \LegendrepolyP{n}@{2x-1}		LegendreP(n, 2*(x) - 1) = LegendreP(n, 2*x - 1)	Error	Successful	Missing Macro Error	-	-
18.7.E13	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}_{2n}\left(1\right)}=\frac{P^{(\alpha,-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,-\frac{1}{2})}_{n}\left(1\right)}}}$ \frac{\JacobipolyP{\alpha}{\alpha}{2n}@{x}}{\JacobipolyP{\alpha}{\alpha}{2n}@{1}} = \frac{\JacobipolyP{\alpha}{-\frac{1}{2}}{n}@{2x^{2}-1}}{\JacobipolyP{\alpha}{-\frac{1}{2}}{n}@{1}}		(JacobiP(2*n, alpha, alpha, x))/(JacobiP(2*n, alpha, alpha, 1)) = (JacobiP(n, alpha, -(1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, -(1)/(2), 1))	Divide[JacobiP[2*n, \[Alpha], \[Alpha], x],JacobiP[2*n, \[Alpha], \[Alpha], 1]] == Divide[JacobiP[n, \[Alpha], -Divide[1,2], 2*(x)^(2)- 1],JacobiP[n, \[Alpha], -Divide[1,2], 1]]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
18.7.E14	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha)}_{2n+1}\left(1\right)}=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,\frac{1}{2})}_{n}\left(1\right)}}}$ \frac{\JacobipolyP{\alpha}{\alpha}{2n+1}@{x}}{\JacobipolyP{\alpha}{\alpha}{2n+1}@{1}} = \frac{x\JacobipolyP{\alpha}{\frac{1}{2}}{n}@{2x^{2}-1}}{\JacobipolyP{\alpha}{\frac{1}{2}}{n}@{1}}		(JacobiP(2*n + 1, alpha, alpha, x))/(JacobiP(2*n + 1, alpha, alpha, 1)) = (x*JacobiP(n, alpha, (1)/(2), 2*(x)^(2)- 1))/(JacobiP(n, alpha, (1)/(2), 1))	Divide[JacobiP[2*n + 1, \[Alpha], \[Alpha], x],JacobiP[2*n + 1, \[Alpha], \[Alpha], 1]] == Divide[x*JacobiP[n, \[Alpha], Divide[1,2], 2*(x)^(2)- 1],JacobiP[n, \[Alpha], Divide[1,2], 1]]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
18.7.E15	${\displaystyle{\displaystyle C^{(\lambda)}_{2n}\left(x\right)=\frac{{\left(\lambda\right)_{n}}}{{\left(\tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}$ \ultrasphpoly{\lambda}{2n}@{x} = \frac{\Pochhammersym{\lambda}{n}}{\Pochhammersym{\tfrac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{-\frac{1}{2}}{n}@{2x^{2}-1}		GegenbauerC(2*n, lambda, x) = (pochhammer(lambda, n))/(pochhammer((1)/(2), n))*JacobiP(n, lambda -(1)/(2), -(1)/(2), 2*(x)^(2)- 1)	GegenbauerC[2*n, \[Lambda], x] == Divide[Pochhammer[\[Lambda], n],Pochhammer[Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], -Divide[1,2], 2*(x)^(2)- 1]	Failure	Failure	Failed [15 / 90] Result: Float(infinity)+Float(infinity)*I Test Values: {lambda = -3/2, x = 3/2, n = 2} Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -3/2, x = 3/2, n = 3} ... skip entries to safe data	Successful [Tested: 90]
18.7.E16	${\displaystyle{\displaystyle C^{(\lambda)}_{2n+1}\left(x\right)=\frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n+1}}}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}$ \ultrasphpoly{\lambda}{2n+1}@{x} = \frac{\Pochhammersym{\lambda}{n+1}}{\Pochhammersym{\frac{1}{2}}{n+1}}x\JacobipolyP{\lambda-\frac{1}{2}}{\frac{1}{2}}{n}@{2x^{2}-1}		GegenbauerC(2*n + 1, lambda, x) = (pochhammer(lambda, n + 1))/(pochhammer((1)/(2), n + 1))*x*JacobiP(n, lambda -(1)/(2), (1)/(2), 2*(x)^(2)- 1)	GegenbauerC[2*n + 1, \[Lambda], x] == Divide[Pochhammer[\[Lambda], n + 1],Pochhammer[Divide[1,2], n + 1]]*x*JacobiP[n, \[Lambda]-Divide[1,2], Divide[1,2], 2*(x)^(2)- 1]	Failure	Failure	Failed [15 / 90] Result: Float(infinity)+Float(infinity)*I Test Values: {lambda = -3/2, x = 3/2, n = 2} Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -3/2, x = 3/2, n = 3} ... skip entries to safe data	Successful [Tested: 90]
18.7.E19	${\displaystyle{\displaystyle H_{2n}\left(x\right)=(-1)^{n}2^{2n}n!L^{(-\frac{1}{2})}_{n}\left(x^{2}\right)}}$ \HermitepolyH{2n}@{x} = (-1)^{n}2^{2n}n!\LaguerrepolyL[-\frac{1}{2}]{n}@{x^{2}}		HermiteH(2*n, x) = (- 1)^(n)* (2)^(2*n)* factorial(n)*LaguerreL(n, -(1)/(2), (x)^(2))	HermiteH[2*n, x] == (- 1)^(n)* (2)^(2*n)* (n)!*LaguerreL[n, -Divide[1,2], (x)^(2)]	Missing Macro Error	Failure	-	Successful [Tested: 9]
18.7.E20	${\displaystyle{\displaystyle H_{2n+1}\left(x\right)=(-1)^{n}2^{2n+1}n!\,xL^{(\frac{1}{2})}_{n}\left(x^{2}\right)}}$ \HermitepolyH{2n+1}@{x} = (-1)^{n}2^{2n+1}n!\,x\LaguerrepolyL[\frac{1}{2}]{n}@{x^{2}}		HermiteH(2*n + 1, x) = (- 1)^(n)* (2)^(2*n + 1)* factorial(n)*x*LaguerreL(n, (1)/(2), (x)^(2))	HermiteH[2*n + 1, x] == (- 1)^(n)* (2)^(2*n + 1)* (n)!*x*LaguerreL[n, Divide[1,2], (x)^(2)]	Missing Macro Error	Failure	-	Successful [Tested: 9]
18.7.E21	${\displaystyle{\displaystyle\lim_{\beta\to\infty}P^{(\alpha,\beta)}_{n}\left(1-(\ifrac{2x}{\beta})\right)=L^{(\alpha)}_{n}\left(x\right)}}$ \lim_{\beta\to\infty}\JacobipolyP{\alpha}{\beta}{n}@{1-(\ifrac{2x}{\beta})} = \LaguerrepolyL[\alpha]{n}@{x}		limit(JacobiP(n, alpha, beta, 1 -((2*x)/(beta))), beta = infinity) = LaguerreL(n, alpha, x)	Limit[JacobiP[n, \[Alpha], \[Beta], 1 -(Divide[2*x,\[Beta]])], \[Beta] -> Infinity, GenerateConditions->None] == LaguerreL[n, \[Alpha], x]	Missing Macro Error	Failure	-	Failed [27 / 27] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[α, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[α, 1.5]} ... skip entries to safe data
18.7.E22	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}P^{(\alpha,\beta)}_{n}\left((2x/\alpha)-1\right)=(-1)^{n}L^{(\beta)}_{n}\left(x\right)}}$ \lim_{\alpha\to\infty}\JacobipolyP{\alpha}{\beta}{n}@{(2x/\alpha)-1} = (-1)^{n}\LaguerrepolyL[\beta]{n}@{x}		limit(JacobiP(n, alpha, beta, (2*x/alpha)- 1), alpha = infinity) = (- 1)^(n)* LaguerreL(n, beta, x)	Limit[JacobiP[n, \[Alpha], \[Beta], (2*x/\[Alpha])- 1], \[Alpha] -> Infinity, GenerateConditions->None] == (- 1)^(n)* LaguerreL[n, \[Beta], x]	Missing Macro Error	Failure	-	Failed [27 / 27] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[β, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[β, 1.5]} ... skip entries to safe data
18.7.E23	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}}}$ \lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}\JacobipolyP{\alpha}{\alpha}{n}@{\alpha^{-\frac{1}{2}}x} = \frac{\HermitepolyH{n}@{x}}{2^{n}n!}		limit((alpha)^(-(1)/(2)*n)* JacobiP(n, alpha, alpha, (alpha)^(-(1)/(2))* x), alpha = infinity) = (HermiteH(n, x))/((2)^(n)* factorial(n))	Limit[\[Alpha]^(-Divide[1,2]*n)* JacobiP[n, \[Alpha], \[Alpha], \[Alpha]^(-Divide[1,2])* x], \[Alpha] -> Infinity, GenerateConditions->None] == Divide[HermiteH[n, x],(2)^(n)* (n)!]	Failure	Aborted	Error	Successful [Tested: 9]
18.7.E24	${\displaystyle{\displaystyle\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}C^{(\lambda)}_{n}\left(\lambda^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}}}$ \lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}\ultrasphpoly{\lambda}{n}@{\lambda^{-\frac{1}{2}}x} = \frac{\HermitepolyH{n}@{x}}{n!}		limit((lambda)^(-(1)/(2)*n)* GegenbauerC(n, lambda, (lambda)^(-(1)/(2))* x), lambda = infinity) = (HermiteH(n, x))/(factorial(n))	Limit[\[Lambda]^(-Divide[1,2]*n)* GegenbauerC[n, \[Lambda], \[Lambda]^(-Divide[1,2])* x], \[Lambda] -> Infinity, GenerateConditions->None] == Divide[HermiteH[n, x],(n)!]	Failure	Aborted	Successful [Tested: 9]	Successful [Tested: 9]
18.7.E25	${\displaystyle{\displaystyle\lim_{\lambda\to 0}\frac{1}{\lambda}C^{(\lambda)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)}}$ \lim_{\lambda\to 0}\frac{1}{\lambda}\ultrasphpoly{\lambda}{n}@{x} = \frac{2}{n}\ChebyshevpolyT{n}@{x}	${\displaystyle{\displaystyle n\geq 1}}$	limit((1)/(lambda)*GegenbauerC(n, lambda, x), lambda = 0) = (2)/(n)*ChebyshevT(n, x)	Limit[Divide[1,\[Lambda]]*GegenbauerC[n, \[Lambda], x], \[Lambda] -> 0, GenerateConditions->None] == Divide[2,n]*ChebyshevT[n, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.7.E26	${\displaystyle{\displaystyle\lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{(\alpha)}_{n}\left((2\alpha)^{\frac{1}{2}}x+\alpha\right)=\frac{(-1)^{n}}{n!}H_{n}\left(x\right)}}$ \lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}\LaguerrepolyL[\alpha]{n}@{(2\alpha)^{\frac{1}{2}}x+\alpha} = \frac{(-1)^{n}}{n!}\HermitepolyH{n}@{x}		limit(((2)/(alpha))^((1)/(2)*n)* LaguerreL(n, alpha, (2*alpha)^((1)/(2))* x + alpha), alpha = infinity) = ((- 1)^(n))/(factorial(n))*HermiteH(n, x)	Limit[(Divide[2,\[Alpha]])^(Divide[1,2]*n)* LaguerreL[n, \[Alpha], (2*\[Alpha])^(Divide[1,2])* x + \[Alpha]], \[Alpha] -> Infinity, GenerateConditions->None] == Divide[(- 1)^(n),(n)!]*HermiteH[n, x]	Missing Macro Error	Aborted	-	Successful [Tested: 9]
18.9#Ex1	${\displaystyle{\displaystyle A_{n}=\dfrac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{2(n+1)(n+\alpha+\beta+1)}}}$ A_{n} = \dfrac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{2(n+1)(n+\alpha+\beta+1)}		A[n] = ((2*n + alpha + beta + 1)*(2*n + alpha + beta + 2))/(2*(n + 1)*(n + alpha + beta + 1))	Subscript[A, n] == Divide[(2*n + \[Alpha]+ \[Beta]+ 1)*(2*n + \[Alpha]+ \[Beta]+ 2),2*(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.9#Ex2	${\displaystyle{\displaystyle B_{n}=\dfrac{(\alpha^{2}-\beta^{2})(2n+\alpha+\beta+1)}{2(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}}}$ B_{n} = \dfrac{(\alpha^{2}-\beta^{2})(2n+\alpha+\beta+1)}{2(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}		B[n] = (((alpha)^(2)- (beta)^(2))*(2*n + alpha + beta + 1))/(2*(n + 1)*(n + alpha + beta + 1)*(2*n + alpha + beta))	Subscript[B, n] == Divide[(\[Alpha]^(2)- \[Beta]^(2))*(2*n + \[Alpha]+ \[Beta]+ 1),2*(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)*(2*n + \[Alpha]+ \[Beta])]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.9#Ex3	${\displaystyle{\displaystyle C_{n}=\dfrac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}}}$ C_{n} = \dfrac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}		C[n] = ((n + alpha)*(n + beta)*(2*n + alpha + beta + 2))/((n + 1)*(n + alpha + beta + 1)*(2*n + alpha + beta))	Subscript[C, n] == Divide[(n + \[Alpha])*(n + \[Beta])*(2*n + \[Alpha]+ \[Beta]+ 2),(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)*(2*n + \[Alpha]+ \[Beta])]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.9.E3	${\displaystyle{\displaystyle P^{(\alpha,\beta-1)}_{n}\left(x\right)-P^{(\alpha-1,\beta)}_{n}\left(x\right)=P^{(\alpha,\beta)}_{n-1}\left(x\right)}}$ \JacobipolyP{\alpha}{\beta-1}{n}@{x}-\JacobipolyP{\alpha-1}{\beta}{n}@{x} = \JacobipolyP{\alpha}{\beta}{n-1}@{x}		JacobiP(n, alpha, beta - 1, x)- JacobiP(n, alpha - 1, beta, x) = JacobiP(n - 1, alpha, beta, x)	JacobiP[n, \[Alpha], \[Beta]- 1, x]- JacobiP[n, \[Alpha]- 1, \[Beta], x] == JacobiP[n - 1, \[Alpha], \[Beta], x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E4	${\displaystyle{\displaystyle(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P^{(\alpha,\beta+1)}_{n}\left(x\right)=2P^{(\alpha,\beta)}_{n}\left(x\right)}}$ (1-x)\JacobipolyP{\alpha+1}{\beta}{n}@{x}+(1+x)\JacobipolyP{\alpha}{\beta+1}{n}@{x} = 2\JacobipolyP{\alpha}{\beta}{n}@{x}		(1 - x)*JacobiP(n, alpha + 1, beta, x)+(1 + x)*JacobiP(n, alpha, beta + 1, x) = 2*JacobiP(n, alpha, beta, x)	(1 - x)*JacobiP[n, \[Alpha]+ 1, \[Beta], x]+(1 + x)*JacobiP[n, \[Alpha], \[Beta]+ 1, x] == 2*JacobiP[n, \[Alpha], \[Beta], x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E5	${\displaystyle{\displaystyle(2n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\alpha+\beta+1)P^{(\alpha,\beta+1)}_{n}\left(x\right)+(n+\alpha)P^{(\alpha,\beta+1)}_{n-1}\left(x\right)}}$ (2n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta}{n}@{x} = (n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta+1}{n}@{x}+(n+\alpha)\JacobipolyP{\alpha}{\beta+1}{n-1}@{x}		(2*n + alpha + beta + 1)*JacobiP(n, alpha, beta, x) = (n + alpha + beta + 1)*JacobiP(n, alpha, beta + 1, x)+(n + alpha)*JacobiP(n - 1, alpha, beta + 1, x)	(2*n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n, \[Alpha], \[Beta], x] == (n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n, \[Alpha], \[Beta]+ 1, x]+(n + \[Alpha])*JacobiP[n - 1, \[Alpha], \[Beta]+ 1, x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E6	${\displaystyle{\displaystyle(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)P^{(\alpha,\beta+1)}_{n}\left(x\right)=(n+1)P^{(\alpha,\beta)}_{n+1}\left(x\right)+(n+\beta+1)P^{(\alpha,\beta)}_{n}\left(x\right)}}$ (n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)\JacobipolyP{\alpha}{\beta+1}{n}@{x} = (n+1)\JacobipolyP{\alpha}{\beta}{n+1}@{x}+(n+\beta+1)\JacobipolyP{\alpha}{\beta}{n}@{x}		(n +(1)/(2)*alpha +(1)/(2)*beta + 1)*(1 + x)*JacobiP(n, alpha, beta + 1, x) = (n + 1)*JacobiP(n + 1, alpha, beta, x)+(n + beta + 1)*JacobiP(n, alpha, beta, x)	(n +Divide[1,2]*\[Alpha]+Divide[1,2]*\[Beta]+ 1)*(1 + x)*JacobiP[n, \[Alpha], \[Beta]+ 1, x] == (n + 1)*JacobiP[n + 1, \[Alpha], \[Beta], x]+(n + \[Beta]+ 1)*JacobiP[n, \[Alpha], \[Beta], x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E7	${\displaystyle{\displaystyle(n+\lambda)C^{(\lambda)}_{n}\left(x\right)=\lambda\left(C^{(\lambda+1)}_{n}\left(x\right)-C^{(\lambda+1)}_{n-2}\left(x\right)\right)}}$ (n+\lambda)\ultrasphpoly{\lambda}{n}@{x} = \lambda\left(\ultrasphpoly{\lambda+1}{n}@{x}-\ultrasphpoly{\lambda+1}{n-2}@{x}\right)		(n + lambda)*GegenbauerC(n, lambda, x) = lambda*(GegenbauerC(n, lambda + 1, x)- GegenbauerC(n - 2, lambda + 1, x))	(n + \[Lambda])*GegenbauerC[n, \[Lambda], x] == \[Lambda]*(GegenbauerC[n, \[Lambda]+ 1, x]- GegenbauerC[n - 2, \[Lambda]+ 1, x])	Successful	Successful	-	Failed [6 / 90] Result: 0.9374999999999998 Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[λ, -1.5]} Result: -0.5 Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[λ, -0.5]} ... skip entries to safe data
18.9.E8	${\displaystyle{\displaystyle 4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}\left(x\right)=-(n+1)(n+2)C^{(\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2\lambda+1)C^{(\lambda)}_{n}\left(x\right)}}$ 4\lambda(n+\lambda+1)(1-x^{2})\ultrasphpoly{\lambda+1}{n}@{x} = -(n+1)(n+2)\ultrasphpoly{\lambda}{n+2}@{x}+(n+2\lambda)(n+2\lambda+1)\ultrasphpoly{\lambda}{n}@{x}		4*lambda*(n + lambda + 1)*(1 - (x)^(2))*GegenbauerC(n, lambda + 1, x) = -(n + 1)*(n + 2)*GegenbauerC(n + 2, lambda, x)+(n + 2*lambda)*(n + 2*lambda + 1)*GegenbauerC(n, lambda, x)	4*\[Lambda]*(n + \[Lambda]+ 1)*(1 - (x)^(2))*GegenbauerC[n, \[Lambda]+ 1, x] == -(n + 1)*(n + 2)*GegenbauerC[n + 2, \[Lambda], x]+(n + 2*\[Lambda])*(n + 2*\[Lambda]+ 1)*GegenbauerC[n, \[Lambda], x]	Successful	Successful	-	Successful [Tested: 90]
18.9.E9	${\displaystyle{\displaystyle T_{n}\left(x\right)=\tfrac{1}{2}\left(U_{n}\left(x\right)-U_{n-2}\left(x\right)\right)}}$ \ChebyshevpolyT{n}@{x} = \tfrac{1}{2}\left(\ChebyshevpolyU{n}@{x}-\ChebyshevpolyU{n-2}@{x}\right)		ChebyshevT(n, x) = (1)/(2)*(ChebyshevU(n, x)- ChebyshevU(n - 2, x))	ChebyshevT[n, x] == Divide[1,2]*(ChebyshevU[n, x]- ChebyshevU[n - 2, x])	Successful	Failure	-	Successful [Tested: 9]
18.9.E10	${\displaystyle{\displaystyle(1-x^{2})U_{n}\left(x\right)=-\tfrac{1}{2}\left(T_{n+2}\left(x\right)-T_{n}\left(x\right)\right)}}$ (1-x^{2})\ChebyshevpolyU{n}@{x} = -\tfrac{1}{2}\left(\ChebyshevpolyT{n+2}@{x}-\ChebyshevpolyT{n}@{x}\right)		(1 - (x)^(2))*ChebyshevU(n, x) = -(1)/(2)*(ChebyshevT(n + 2, x)- ChebyshevT(n, x))	(1 - (x)^(2))*ChebyshevU[n, x] == -Divide[1,2]*(ChebyshevT[n + 2, x]- ChebyshevT[n, x])	Successful	Failure	-	Successful [Tested: 9]
18.9.E13	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=L^{(\alpha+1)}_{n}\left(x\right)-L^{(\alpha+1)}_{n-1}\left(x\right)}}$ \LaguerrepolyL[\alpha]{n}@{x} = \LaguerrepolyL[\alpha+1]{n}@{x}-\LaguerrepolyL[\alpha+1]{n-1}@{x}		LaguerreL(n, alpha, x) = LaguerreL(n, alpha + 1, x)- LaguerreL(n - 1, alpha + 1, x)	LaguerreL[n, \[Alpha], x] == LaguerreL[n, \[Alpha]+ 1, x]- LaguerreL[n - 1, \[Alpha]+ 1, x]	Missing Macro Error	Successful	-	Successful [Tested: 27]
18.9.E14	${\displaystyle{\displaystyle xL^{(\alpha+1)}_{n}\left(x\right)=-(n+1)L^{(\alpha)}_{n+1}\left(x\right)+(n+\alpha+1)L^{(\alpha)}_{n}\left(x\right)}}$ x\LaguerrepolyL[\alpha+1]{n}@{x} = -(n+1)\LaguerrepolyL[\alpha]{n+1}@{x}+(n+\alpha+1)\LaguerrepolyL[\alpha]{n}@{x}		x*LaguerreL(n, alpha + 1, x) = -(n + 1)*LaguerreL(n + 1, alpha, x)+(n + alpha + 1)*LaguerreL(n, alpha, x)	x*LaguerreL[n, \[Alpha]+ 1, x] == -(n + 1)*LaguerreL[n + 1, \[Alpha], x]+(n + \[Alpha]+ 1)*LaguerreL[n, \[Alpha], x]	Missing Macro Error	Successful	-	Successful [Tested: 27]
18.9.E15	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=\tfrac{1}{2}(n+\alpha+\beta+1)P^{(\alpha+1,\beta+1)}_{n-1}\left(x\right)}}$ \deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = \tfrac{1}{2}(n+\alpha+\beta+1)\JacobipolyP{\alpha+1}{\beta+1}{n-1}@{x}		diff(JacobiP(n, alpha, beta, x), x) = (1)/(2)*(n + alpha + beta + 1)*JacobiP(n - 1, alpha + 1, beta + 1, x)	D[JacobiP[n, \[Alpha], \[Beta], x], x] == Divide[1,2]*(n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n - 1, \[Alpha]+ 1, \[Beta]+ 1, x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E16	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)\right)=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P^{(\alpha-1,\beta-1)}_{n+1}\left(x\right)}}$ \deriv{}{x}\left((1-x)^{\alpha}(1+x)^{\beta}\JacobipolyP{\alpha}{\beta}{n}@{x}\right) = -2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}\JacobipolyP{\alpha-1}{\beta-1}{n+1}@{x}		diff((1 - x)^(alpha)*(1 + x)^(beta)* JacobiP(n, alpha, beta, x), x) = - 2*(n + 1)*(1 - x)^(alpha - 1)*(1 + x)^(beta - 1)* JacobiP(n + 1, alpha - 1, beta - 1, x)	D[(1 - x)^\[Alpha]*(1 + x)^\[Beta]* JacobiP[n, \[Alpha], \[Beta], x], x] == - 2*(n + 1)*(1 - x)^(\[Alpha]- 1)*(1 + x)^(\[Beta]- 1)* JacobiP[n + 1, \[Alpha]- 1, \[Beta]- 1, x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E17	${\displaystyle{\displaystyle(2n+\alpha+\beta)(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=n\left(\alpha-\beta-(2n+\alpha+\beta)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)+2(n+\alpha)(n+\beta)P^{(\alpha,\beta)}_{n-1}\left(x\right)}}$ (2n+\alpha+\beta)(1-x^{2})\deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = n\left(\alpha-\beta-(2n+\alpha+\beta)x\right)\JacobipolyP{\alpha}{\beta}{n}@{x}+2(n+\alpha)(n+\beta)\JacobipolyP{\alpha}{\beta}{n-1}@{x}		(2*n + alpha + beta)*(1 - (x)^(2))*diff(JacobiP(n, alpha, beta, x), x) = n*(alpha - beta -(2*n + alpha + beta)*x)*JacobiP(n, alpha, beta, x)+ 2*(n + alpha)*(n + beta)*JacobiP(n - 1, alpha, beta, x)	(2*n + \[Alpha]+ \[Beta])*(1 - (x)^(2))*D[JacobiP[n, \[Alpha], \[Beta], x], x] == n*(\[Alpha]- \[Beta]-(2*n + \[Alpha]+ \[Beta])*x)*JacobiP[n, \[Alpha], \[Beta], x]+ 2*(n + \[Alpha])*(n + \[Beta])*JacobiP[n - 1, \[Alpha], \[Beta], x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E18	${\displaystyle{\displaystyle(2n+\alpha+\beta+2)(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\alpha+\beta+1)\left(\alpha-\beta+(2n+\alpha+\beta+2)x\right)P^{(\alpha,\beta)}_{n}\left(x\right)-2(n+1)(n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n+1}\left(x\right)}}$ (2n+\alpha+\beta+2)(1-x^{2})\deriv{}{x}\JacobipolyP{\alpha}{\beta}{n}@{x} = (n+\alpha+\beta+1)\left(\alpha-\beta+(2n+\alpha+\beta+2)x\right)\JacobipolyP{\alpha}{\beta}{n}@{x}-2(n+1)(n+\alpha+\beta+1)\JacobipolyP{\alpha}{\beta}{n+1}@{x}		(2*n + alpha + beta + 2)*(1 - (x)^(2))*diff(JacobiP(n, alpha, beta, x), x) = (n + alpha + beta + 1)*(alpha - beta +(2*n + alpha + beta + 2)*x)*JacobiP(n, alpha, beta, x)- 2*(n + 1)*(n + alpha + beta + 1)*JacobiP(n + 1, alpha, beta, x)	(2*n + \[Alpha]+ \[Beta]+ 2)*(1 - (x)^(2))*D[JacobiP[n, \[Alpha], \[Beta], x], x] == (n + \[Alpha]+ \[Beta]+ 1)*(\[Alpha]- \[Beta]+(2*n + \[Alpha]+ \[Beta]+ 2)*x)*JacobiP[n, \[Alpha], \[Beta], x]- 2*(n + 1)*(n + \[Alpha]+ \[Beta]+ 1)*JacobiP[n + 1, \[Alpha], \[Beta], x]	Failure	Successful	Successful [Tested: 81]	Successful [Tested: 81]
18.9.E19	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}\left(x\right)=2\lambda C^{(\lambda+1)}_{n-1}\left(x\right)}}$ \deriv{}{x}\ultrasphpoly{\lambda}{n}@{x} = 2\lambda\ultrasphpoly{\lambda+1}{n-1}@{x}		diff(GegenbauerC(n, lambda, x), x) = 2*lambda*GegenbauerC(n - 1, lambda + 1, x)	D[GegenbauerC[n, \[Lambda], x], x] == 2*\[Lambda]*GegenbauerC[n - 1, \[Lambda]+ 1, x]	Successful	Successful	-	Successful [Tested: 90]
18.9.E20	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)\right)=-\frac{(n+1)(n+2\lambda-1)}{2(\lambda-1)}{(1-x^{2})^{\lambda-\frac{3}{2}}}C^{(\lambda-1)}_{n+1}\left(x\right)}}$ \deriv{}{x}\left((1-x^{2})^{\lambda-\frac{1}{2}}\ultrasphpoly{\lambda}{n}@{x}\right) = -\frac{(n+1)(n+2\lambda-1)}{2(\lambda-1)}{(1-x^{2})^{\lambda-\frac{3}{2}}}\ultrasphpoly{\lambda-1}{n+1}@{x}		diff((1 - (x)^(2))^(lambda -(1)/(2))* GegenbauerC(n, lambda, x), x) = -((n + 1)*(n + 2*lambda - 1))/(2*(lambda - 1))*(1 - (x)^(2))^(lambda -(3)/(2))*GegenbauerC(n + 1, lambda - 1, x)	D[(1 - (x)^(2))^(\[Lambda]-Divide[1,2])* GegenbauerC[n, \[Lambda], x], x] == -Divide[(n + 1)*(n + 2*\[Lambda]- 1),2*(\[Lambda]- 1)]*(1 - (x)^(2))^(\[Lambda]-Divide[3,2])*GegenbauerC[n + 1, \[Lambda]- 1, x]	Successful	Successful	-	Successful [Tested: 90]
18.9.E21	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}T_{n}\left(x\right)=nU_{n-1}\left(x\right)}}$ \deriv{}{x}\ChebyshevpolyT{n}@{x} = n\ChebyshevpolyU{n-1}@{x}		diff(ChebyshevT(n, x), x) = n*ChebyshevU(n - 1, x)	D[ChebyshevT[n, x], x] == n*ChebyshevU[n - 1, x]	Successful	Successful	-	Successful [Tested: 9]
18.9.E22	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{\frac{1}{2}}U_{n}\left(x\right)\right)=-(n+1){(1-x^{2})^{-\frac{1}{2}}}T_{n+1}\left(x\right)}}$ \deriv{}{x}\left((1-x^{2})^{\frac{1}{2}}\ChebyshevpolyU{n}@{x}\right) = -(n+1){(1-x^{2})^{-\frac{1}{2}}}\ChebyshevpolyT{n+1}@{x}		diff((1 - (x)^(2))^((1)/(2))* ChebyshevU(n, x), x) = -(n + 1)*(1 - (x)^(2))^(-(1)/(2))*ChebyshevT(n + 1, x)	D[(1 - (x)^(2))^(Divide[1,2])* ChebyshevU[n, x], x] == -(n + 1)*(1 - (x)^(2))^(-Divide[1,2])*ChebyshevT[n + 1, x]	Successful	Successful	-	Successful [Tested: 9]
18.9.E23	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}L^{(\alpha)}_{n}\left(x\right)=-L^{(\alpha+1)}_{n-1}\left(x\right)}}$ \deriv{}{x}\LaguerrepolyL[\alpha]{n}@{x} = -\LaguerrepolyL[\alpha+1]{n-1}@{x}		diff(LaguerreL(n, alpha, x), x) = - LaguerreL(n - 1, alpha + 1, x)	D[LaguerreL[n, \[Alpha], x], x] == - LaguerreL[n - 1, \[Alpha]+ 1, x]	Missing Macro Error	Successful	-	Successful [Tested: 27]
18.9.E24	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}x^{\alpha}L^{(\alpha)}_{n}\left(x\right)\right)=(n+1)e^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}\left(x\right)}}$ \deriv{}{x}\left(e^{-x}x^{\alpha}\LaguerrepolyL[\alpha]{n}@{x}\right) = (n+1)e^{-x}x^{\alpha-1}\LaguerrepolyL[\alpha-1]{n+1}@{x}		diff(exp(- x)*(x)^(alpha)* LaguerreL(n, alpha, x), x) = (n + 1)*exp(- x)*(x)^(alpha - 1)* LaguerreL(n + 1, alpha - 1, x)	D[Exp[- x]*(x)^\[Alpha]* LaguerreL[n, \[Alpha], x], x] == (n + 1)*Exp[- x]*(x)^(\[Alpha]- 1)* LaguerreL[n + 1, \[Alpha]- 1, x]	Missing Macro Error	Successful	-	Successful [Tested: 27]
18.9.E25	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}H_{n}\left(x\right)=2nH_{n-1}\left(x\right)}}$ \deriv{}{x}\HermitepolyH{n}@{x} = 2n\HermitepolyH{n-1}@{x}		diff(HermiteH(n, x), x) = 2*n*HermiteH(n - 1, x)	D[HermiteH[n, x], x] == 2*n*HermiteH[n - 1, x]	Successful	Successful	-	Successful [Tested: 9]
18.9.E26	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x^{2}}H_{n}\left(x\right)\right)=-e^{-x^{2}}H_{n+1}\left(x\right)}}$ \deriv{}{x}\left(e^{-x^{2}}\HermitepolyH{n}@{x}\right) = -e^{-x^{2}}\HermitepolyH{n+1}@{x}		diff(exp(- (x)^(2))*HermiteH(n, x), x) = - exp(- (x)^(2))*HermiteH(n + 1, x)	D[Exp[- (x)^(2)]*HermiteH[n, x], x] == - Exp[- (x)^(2)]*HermiteH[n + 1, x]	Successful	Successful	-	Successful [Tested: 9]
18.10.E1	${\displaystyle{\displaystyle\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}}}$ \frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}} = \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi,\alpha>-\tfrac{1}{2}}}$	(JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1)) = (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1))	Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]] == Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 27]
18.10.E1	${\displaystyle{\displaystyle\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2}}\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma\left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}}}\mathrm{d}\phi}}$ \frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}} = \frac{2^{\alpha+\frac{1}{2}}\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma@{\alpha+\frac{1}{2}}}(\sin@@{\theta})^{-2\alpha}\int_{0}^{\theta}\frac{\cos@{(n+\alpha+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{-\alpha+\frac{1}{2}}}\diff{\phi}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi,\alpha>-\tfrac{1}{2},\Re(\alpha+1)>0,\Re(\alpha+\frac{1}{2})>0}}$	(GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1)) = ((2)^(alpha +(1)/(2))* GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))*(sin(theta))^(- 2*alpha)* int((cos((n + alpha +(1)/(2))*phi))/((cos(phi)- cos(theta))^(- alpha +(1)/(2))), phi = 0..theta)	Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]] == Divide[(2)^(\[Alpha]+Divide[1,2])* Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]]*(Sin[\[Theta]])^(- 2*\[Alpha])* Integrate[Divide[Cos[(n + \[Alpha]+Divide[1,2])*\[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(- \[Alpha]+Divide[1,2])], {\[Phi], 0, \[Theta]}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 27]	Skipped - Because timed out
18.10.E2	${\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac{\cos\left((n+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{\frac{1}{2}}}\mathrm{d}\phi}}$ \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac{\cos@{(n+\tfrac{1}{2})\phi}}{(\cos@@{\phi}-\cos@@{\theta})^{\frac{1}{2}}}\diff{\phi}	${\displaystyle{\displaystyle 0<\theta,\theta<\pi}}$	LegendreP(n, cos(theta)) = ((2)^((1)/(2)))/(Pi)*int((cos((n +(1)/(2))*phi))/((cos(phi)- cos(theta))^((1)/(2))), phi = 0..theta)	LegendreP[n, Cos[\[Theta]]] == Divide[(2)^(Divide[1,2]),Pi]*Integrate[Divide[Cos[(n +Divide[1,2])*\[Phi]],(Cos[\[Phi]]- Cos[\[Theta]])^(Divide[1,2])], {\[Phi], 0, \[Theta]}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Skipped - Because timed out
18.10.E4	${\displaystyle{\displaystyle{\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}}=\frac{\Gamma\left(\alpha+1\right)}{\pi^{\frac{1}{2}}\Gamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos\theta+i\sin\theta\cos\phi)^{n}\*(\sin\phi)^{2\alpha}\mathrm{d}\phi}}}$ {\frac{\JacobipolyP{\alpha}{\alpha}{n}@{\cos@@{\theta}}}{\JacobipolyP{\alpha}{\alpha}{n}@{1}}=\frac{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{\cos@@{\theta}}}{\ultrasphpoly{\alpha+\frac{1}{2}}{n}@{1}}} = \frac{\EulerGamma@{\alpha+1}}{\pi^{\frac{1}{2}}\EulerGamma{(\alpha+\tfrac{1}{2})}}\*{\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\*(\sin@@{\phi})^{2\alpha}\diff{\phi}}	${\displaystyle{\displaystyle\alpha>-\frac{1}{2},\Re(\alpha+1)>0}}$	(JacobiP(n, alpha, alpha, cos(theta)))/(JacobiP(n, alpha, alpha, 1)) = (GegenbauerC(n, alpha +(1)/(2), cos(theta)))/(GegenbauerC(n, alpha +(1)/(2), 1)) = (GAMMA(alpha + 1))/((Pi)^((1)/(2))* GAMMA(alpha +(1)/(2)))*int((cos(theta)+ I*sin(theta)*cos(phi))^(n)*(sin(phi))^(2*alpha), phi = 0..Pi)	Divide[JacobiP[n, \[Alpha], \[Alpha], Cos[\[Theta]]],JacobiP[n, \[Alpha], \[Alpha], 1]] == Divide[GegenbauerC[n, \[Alpha]+Divide[1,2], Cos[\[Theta]]],GegenbauerC[n, \[Alpha]+Divide[1,2], 1]] == Divide[Gamma[\[Alpha]+ 1],(Pi)^(Divide[1,2])* Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cos[\[Phi]])^(n)*(Sin[\[Phi]])^(2*\[Alpha]), {\[Phi], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
18.10.E5	${\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0}^{\pi}(\cos\theta+i\sin\theta\cos\phi)^{n}\mathrm{d}\phi}}$ \LegendrepolyP{n}@{\cos@@{\theta}} = \frac{1}{\pi}\int_{0}^{\pi}(\cos@@{\theta}+i\sin@@{\theta}\cos@@{\phi})^{n}\diff{\phi}		LegendreP(n, cos(theta)) = (1)/(Pi)*int((cos(theta)+ I*sin(theta)*cos(phi))^(n), phi = 0..Pi)	LegendreP[n, Cos[\[Theta]]] == Divide[1,Pi]*Integrate[(Cos[\[Theta]]+ I*Sin[\[Theta]]*Cos[\[Phi]])^(n), {\[Phi], 0, Pi}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 30]	Skipped - Because timed out
18.10.E7	${\displaystyle{\displaystyle H_{n}\left(x\right)=\frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\mathrm{d}t}}$ \HermitepolyH{n}@{x} = \frac{2^{n}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}(x+it)^{n}e^{-t^{2}}\diff{t}		HermiteH(n, x) = ((2)^(n))/((Pi)^((1)/(2)))*int((x + I*t)^(n)* exp(- (t)^(2)), t = - infinity..infinity)	HermiteH[n, x] == Divide[(2)^(n),(Pi)^(Divide[1,2])]*Integrate[(x + I*t)^(n)* Exp[- (t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 9]	Failed [9 / 9] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
18.10.E9	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\frac{e^{x}x^{-\frac{1}{2}\alpha}}{n!}\int_{0}^{\infty}e^{-t}t^{n+\frac{1}{2}\alpha}J_{\alpha}\left(2\sqrt{xt}\right)\mathrm{d}t}}$ \LaguerrepolyL[\alpha]{n}@{x} = \frac{e^{x}x^{-\frac{1}{2}\alpha}}{n!}\int_{0}^{\infty}e^{-t}t^{n+\frac{1}{2}\alpha}\BesselJ{\alpha}@{2\sqrt{xt}}\diff{t}	${\displaystyle{\displaystyle\alpha>-1,\Re((\alpha)+k+1)>0}}$	LaguerreL(n, alpha, x) = (exp(x)*(x)^(-(1)/(2)*alpha))/(factorial(n))*int(exp(- t)*(t)^(n +(1)/(2)*alpha)* BesselJ(alpha, 2*sqrt(x*t)), t = 0..infinity)	LaguerreL[n, \[Alpha], x] == Divide[Exp[x]*(x)^(-Divide[1,2]*\[Alpha]),(n)!]*Integrate[Exp[- t]*(t)^(n +Divide[1,2]*\[Alpha])* BesselJ[\[Alpha], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
18.10.E10	${\displaystyle{\displaystyle H_{n}\left(x\right)=\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t}}$ \HermitepolyH{n}@{x} = \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t}		HermiteH(n, x) = ((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity)	HermiteH[n, x] == Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.10.E10	${\displaystyle{\displaystyle\frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\mathrm{d}t=\frac{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos\left(2xt-\tfrac{1}{2}n\pi\right)\mathrm{d}t}}$ \frac{(-2i)^{n}e^{x^{2}}}{\pi^{\frac{1}{2}}}\int_{-\infty}^{\infty}e^{-t^{2}}t^{n}e^{2ixt}\diff{t} = \frac{2^{n+1}}{\pi^{\frac{1}{2}}}e^{x^{2}}\int_{0}^{\infty}e^{-t^{2}}t^{n}\cos@{2xt-\tfrac{1}{2}n\pi}\diff{t}		((- 2*I)^(n)* exp((x)^(2)))/((Pi)^((1)/(2)))*int(exp(- (t)^(2))*(t)^(n)* exp(2*I*x*t), t = - infinity..infinity) = ((2)^(n + 1))/((Pi)^((1)/(2)))*exp((x)^(2))*int(exp(- (t)^(2))*(t)^(n)* cos(2*x*t -(1)/(2)*n*Pi), t = 0..infinity)	Divide[(- 2*I)^(n)* Exp[(x)^(2)],(Pi)^(Divide[1,2])]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Exp[2*I*x*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(2)^(n + 1),(Pi)^(Divide[1,2])]*Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)]*(t)^(n)* Cos[2*x*t -Divide[1,2]*n*Pi], {t, 0, Infinity}, GenerateConditions->None]	Failure	Aborted	Successful [Tested: 9]	Successful [Tested: 9]
18.11.E1	${\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)}}$ \FerrersP[m]{n}@{x} = \Pochhammersym{\tfrac{1}{2}}{m}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}\ultrasphpoly{m+\frac{1}{2}}{n-m}@{x}	${\displaystyle{\displaystyle 0\leq m,m\leq n,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	LegendreP(n, m, x) = pochhammer((1)/(2), m)*(- 2)^(m)*(1 - (x)^(2))^((1)/(2)*m)* GegenbauerC(n - m, m +(1)/(2), x)	LegendreP[n, m, x] == Pochhammer[Divide[1,2], m]*(- 2)^(m)*(1 - (x)^(2))^(Divide[1,2]*m)* GegenbauerC[n - m, m +Divide[1,2], x]	Failure	Failure	Successful [Tested: 18]	Successful [Tested: 18]
18.11.E1	${\displaystyle{\displaystyle{\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m}}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right)}}$ \Pochhammersym{\tfrac{1}{2}}{m}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}\ultrasphpoly{m+\frac{1}{2}}{n-m}@{x} = \Pochhammersym{n+1}{m}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}\JacobipolyP{m}{m}{n-m}@{x}	${\displaystyle{\displaystyle 0\leq m,m\leq n,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}}$	pochhammer((1)/(2), m)*(- 2)^(m)*(1 - (x)^(2))^((1)/(2)*m)* GegenbauerC(n - m, m +(1)/(2), x) = pochhammer(n + 1, m)*(- 2)^(- m)*(1 - (x)^(2))^((1)/(2)*m)* JacobiP(n - m, m, m, x)	Pochhammer[Divide[1,2], m]*(- 2)^(m)*(1 - (x)^(2))^(Divide[1,2]*m)* GegenbauerC[n - m, m +Divide[1,2], x] == Pochhammer[n + 1, m]*(- 2)^(- m)*(1 - (x)^(2))^(Divide[1,2]*m)* JacobiP[n - m, m, m, x]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 18]
18.11.E2	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}M\left(-n,\alpha+1,x\right)}}$ \LaguerrepolyL[\alpha]{n}@{x} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\KummerconfhyperM@{-n}{\alpha+1}{x}		LaguerreL(n, alpha, x) = (pochhammer(alpha + 1, n))/(factorial(n))*KummerM(- n, alpha + 1, x)	LaguerreL[n, \[Alpha], x] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric1F1[- n, \[Alpha]+ 1, x]	Missing Macro Error	Successful	Skip - symbolical successful subtest	Successful [Tested: 27]
18.11.E2	${\displaystyle{\displaystyle\frac{{\left(\alpha+1\right)_{n}}}{n!}M\left(-n,\alpha+1,x\right)=\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)}}$ \frac{\Pochhammersym{\alpha+1}{n}}{n!}\KummerconfhyperM@{-n}{\alpha+1}{x} = \frac{(-1)^{n}}{n!}\KummerconfhyperU@{-n}{\alpha+1}{x}		(pochhammer(alpha + 1, n))/(factorial(n))*KummerM(- n, alpha + 1, x) = ((- 1)^(n))/(factorial(n))*KummerU(- n, alpha + 1, x)	Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric1F1[- n, \[Alpha]+ 1, x] == Divide[(- 1)^(n),(n)!]*HypergeometricU[- n, \[Alpha]+ 1, x]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
18.11.E2	${\displaystyle{\displaystyle\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)}}$ \frac{(-1)^{n}}{n!}\KummerconfhyperU@{-n}{\alpha+1}{x} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperM{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x}		((- 1)^(n))/(factorial(n))*KummerU(- n, alpha + 1, x) = (pochhammer(alpha + 1, n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerM(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x)	Divide[(- 1)^(n),(n)!]*HypergeometricU[- n, \[Alpha]+ 1, x] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerM[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
18.11.E2	${\displaystyle{\displaystyle\frac{{\left(\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)=\frac{(-1)^{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}W_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)}}$ \frac{\Pochhammersym{\alpha+1}{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperM{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x} = \frac{(-1)^{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\WhittakerconfhyperW{n+\frac{1}{2}(\alpha+1)}{\frac{1}{2}\alpha}@{x}		(pochhammer(alpha + 1, n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerM(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x) = ((- 1)^(n))/(factorial(n))*(x)^(-(1)/(2)*(alpha + 1))* exp((1)/(2)*x)*WhittakerW(n +(1)/(2)*(alpha + 1), (1)/(2)*alpha, x)	Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerM[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x] == Divide[(- 1)^(n),(n)!]*(x)^(-Divide[1,2]*(\[Alpha]+ 1))* Exp[Divide[1,2]*x]*WhittakerW[n +Divide[1,2]*(\[Alpha]+ 1), Divide[1,2]*\[Alpha], x]	Failure	Failure	Successful [Tested: 27]	Successful [Tested: 27]
18.11.E3	${\displaystyle{\displaystyle H_{n}\left(x\right)=2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}\right)}}$ \HermitepolyH{n}@{x} = 2^{n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{x^{2}}		HermiteH(n, x) = (2)^(n)* KummerU(-(1)/(2)*n, (1)/(2), (x)^(2))	HermiteH[n, x] == (2)^(n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], (x)^(2)]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.11.E3	${\displaystyle{\displaystyle 2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}\right)=2^{n}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)}}$ 2^{n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{x^{2}} = 2^{n}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{x^{2}}		(2)^(n)* KummerU(-(1)/(2)*n, (1)/(2), (x)^(2)) = (2)^(n)* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (x)^(2))	(2)^(n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], (x)^(2)] == (2)^(n)* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], (x)^(2)]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.11.E3	${\displaystyle{\displaystyle 2^{n}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)=2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)}}$ 2^{n}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{x^{2}} = 2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}\paraU@{-n-\tfrac{1}{2}}{2^{\frac{1}{2}}x}		(2)^(n)* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (x)^(2)) = (2)^((1)/(2)*n)* exp((1)/(2)*(x)^(2))*CylinderU(- n -(1)/(2), (2)^((1)/(2))* x)	(2)^(n)* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], (x)^(2)] == (2)^(Divide[1,2]*n)* Exp[Divide[1,2]*(x)^(2)]*ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), (2)^(Divide[1,2])* x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.11.E4	${\displaystyle{\displaystyle 2^{\frac{1}{2}n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},\tfrac{1}{2}x^{2}\right)=2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)}}$ 2^{\frac{1}{2}n}\KummerconfhyperU@{-\tfrac{1}{2}n}{\tfrac{1}{2}}{\tfrac{1}{2}x^{2}} = 2^{\frac{1}{2}(n-1)}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}}		(2)^((1)/(2)*n)* KummerU(-(1)/(2)*n, (1)/(2), (1)/(2)*(x)^(2)) = (2)^((1)/(2)*(n - 1))* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (1)/(2)*(x)^(2))	(2)^(Divide[1,2]*n)* HypergeometricU[-Divide[1,2]*n, Divide[1,2], Divide[1,2]*(x)^(2)] == (2)^(Divide[1,2]*(n - 1))* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], Divide[1,2]*(x)^(2)]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.11.E4	${\displaystyle{\displaystyle 2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)=e^{\tfrac{1}{4}x^{2}}U\left(-n-\tfrac{1}{2},x\right)}}$ 2^{\frac{1}{2}(n-1)}x\KummerconfhyperU@{-\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}} = e^{\tfrac{1}{4}x^{2}}\paraU@{-n-\tfrac{1}{2}}{x}		(2)^((1)/(2)*(n - 1))* x*KummerU(-(1)/(2)*n +(1)/(2), (3)/(2), (1)/(2)*(x)^(2)) = exp((1)/(4)*(x)^(2))*CylinderU(- n -(1)/(2), x)	(2)^(Divide[1,2]*(n - 1))* x*HypergeometricU[-Divide[1,2]*n +Divide[1,2], Divide[3,2], Divide[1,2]*(x)^(2)] == Exp[Divide[1,4]*(x)^(2)]*ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.11.E5	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(\cos\frac{z}{n}\right)}}$ \lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{1-\frac{z^{2}}{2n^{2}}} = \lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\frac{z}{n}}}		limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, 1 -((z)^(2))/(2*(n)^(2))), n = infinity) = limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity)	Limit[Divide[1,(n)^\[Alpha]]*JacobiP[n, \[Alpha], \[Beta], 1 -Divide[(z)^(2),2*(n)^(2)]], n -> Infinity, GenerateConditions->None] == Limit[Divide[1,(n)^\[Alpha]]*JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity, GenerateConditions->None]	Failure	Aborted	Error	Skipped - Because timed out
18.11.E5	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha}\left(z\right)}}$ \lim_{n\to\infty}\frac{1}{n^{\alpha}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\frac{z}{n}}} = \frac{2^{\alpha}}{z^{\alpha}}\BesselJ{\alpha}@{z}	${\displaystyle{\displaystyle\Re((\alpha)+k+1)>0}}$	limit((1)/((n)^(alpha))*JacobiP(n, alpha, beta, cos((z)/(n))), n = infinity) = ((2)^(alpha))/((z)^(alpha))*BesselJ(alpha, z)	Limit[Divide[1,(n)^\[Alpha]]*JacobiP[n, \[Alpha], \[Beta], Cos[Divide[z,n]]], n -> Infinity, GenerateConditions->None] == Divide[(2)^\[Alpha],(z)^\[Alpha]]*BesselJ[\[Alpha], z]	Failure	Aborted	Error	Skipped - Because timed out
18.11.E6	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{1}{n^{\alpha}}L^{(\alpha)}_{n}\left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha}}J_{\alpha}\left(2z^{\frac{1}{2}}\right)}}$ \lim_{n\to\infty}\frac{1}{n^{\alpha}}\LaguerrepolyL[\alpha]{n}@{\frac{z}{n}} = \frac{1}{z^{\frac{1}{2}\alpha}}\BesselJ{\alpha}@{2z^{\frac{1}{2}}}	${\displaystyle{\displaystyle\Re((\alpha)+k+1)>0}}$	limit((1)/((n)^(alpha))*LaguerreL(n, alpha, (z)/(n)), n = infinity) = (1)/((z)^((1)/(2)*alpha))*BesselJ(alpha, 2*(z)^((1)/(2)))	Limit[Divide[1,(n)^\[Alpha]]*LaguerreL[n, \[Alpha], Divide[z,n]], n -> Infinity, GenerateConditions->None] == Divide[1,(z)^(Divide[1,2]*\[Alpha])]*BesselJ[\[Alpha], 2*(z)^(Divide[1,2])]	Missing Macro Error	Aborted	-	Failed [21 / 21] Result: Plus[Complex[-0.5130891006146308, 0.11628471920726866], Limit[Times[Power[n, -1.5], LaguerreL[n, 1.5, Times[Complex[0.8660254037844387, 0.49999999999999994], Power[n, -1]]]], Rule[n, DirectedInfinity[1]], Rule[GenerateConditions, None]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Plus[Complex[-0.5517607501957961, 0.2594860904083832], Limit[Times[Power[n, -0.5], LaguerreL[n, 0.5, Times[Complex[0.8660254037844387, 0.49999999999999994], Power[n, -1]]]], Rule[n, DirectedInfinity[1]], Rule[GenerateConditions, None]]] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
18.11.E7	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^{2n}n!}H_{2n}\left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{1}{\pi^{\frac{1}{2}}}\cos z}}$ \lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^{2n}n!}\HermitepolyH{2n}@{\frac{z}{2n^{\frac{1}{2}}}} = \frac{1}{\pi^{\frac{1}{2}}}\cos@@{z}		limit(((- 1)^(n)* (n)^((1)/(2)))/((2)^(2*n)* factorial(n))*HermiteH(2*n, (z)/(2*(n)^((1)/(2)))), n = infinity) = (1)/((Pi)^((1)/(2)))*cos(z)	Limit[Divide[(- 1)^(n)* (n)^(Divide[1,2]),(2)^(2*n)* (n)!]*HermiteH[2*n, Divide[z,2*(n)^(Divide[1,2])]], n -> Infinity, GenerateConditions->None] == Divide[1,(Pi)^(Divide[1,2])]*Cos[z]	Failure	Aborted	Successful [Tested: 7]	Skipped - Because timed out
18.11.E8	${\displaystyle{\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}H_{2n+1}\left(\frac{z}{2n^{\frac{1}{2}}}\right)=\frac{2}{\pi^{\frac{1}{2}}}\sin z}}$ \lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}\HermitepolyH{2n+1}@{\frac{z}{2n^{\frac{1}{2}}}} = \frac{2}{\pi^{\frac{1}{2}}}\sin@@{z}		limit(((- 1)^(n))/((2)^(2*n)* factorial(n))*HermiteH(2*n + 1, (z)/(2*(n)^((1)/(2)))), n = infinity) = (2)/((Pi)^((1)/(2)))*sin(z)	Limit[Divide[(- 1)^(n),(2)^(2*n)* (n)!]*HermiteH[2*n + 1, Divide[z,2*(n)^(Divide[1,2])]], n -> Infinity, GenerateConditions->None] == Divide[2,(Pi)^(Divide[1,2])]*Sin[z]	Failure	Aborted	Error	Skipped - Because timed out
18.12.E1	${\displaystyle{\displaystyle\frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{\beta}}=\sum_{n=0}^{\infty}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n}}}$ \frac{2^{\alpha+\beta}}{R(1+R-z)^{\alpha}(1+R+z)^{\beta}} = \sum_{n=0}^{\infty}\JacobipolyP{\alpha}{\beta}{n}@{x}z^{n}	${\displaystyle{\displaystyle R=\sqrt{1-2xz+z^{2}},|z|<1}}$	((2)^(alpha + beta))/(R*(1 + R -(x + y*I))^(alpha)*(1 + R +(x + y*I))^(beta)) = sum(JacobiP(n, alpha, beta, x)*(x + y*I)^(n), n = 0..infinity)	Divide[(2)^(\[Alpha]+ \[Beta]),R*(1 + R -(x + y*I))^\[Alpha]*(1 + R +(x + y*I))^\[Beta]] == Sum[JacobiP[n, \[Alpha], \[Beta], x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Manual Skip!	Failed [300 / 300] Result: Plus[Complex[-0.23827892567037992, -0.3450900635900643], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], JacobiP[n, 1.5, 1.5, 1.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[R, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5]} Result: Plus[Complex[-0.5735714902915137, -0.46165149748368195], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], JacobiP[n, 1.5, 0.5, 1.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[R, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 0.5]} ... skip entries to safe data
18.12.E2	${\displaystyle{\displaystyle\left(\tfrac{1}{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\*\left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{\Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n}}}$ \left(\tfrac{1}{2}(1-x)z\right)^{-\frac{1}{2}\alpha}\BesselJ{\alpha}@{\sqrt{2(1-x)z}}\*\left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}\modBesselI{\beta}@{\sqrt{2(1+x)z}} = \sum_{n=0}^{\infty}\frac{\JacobipolyP{\alpha}{\beta}{n}@{x}}{\EulerGamma@{n+\alpha+1}\EulerGamma@{n+\beta+1}}z^{n}	${\displaystyle{\displaystyle\Re((\alpha)+k+1)>0,\Re(n+\alpha+1)>0,\Re(n+\beta+1)>0,\Re((\beta)+k+1)>0}}$	((1)/(2)*(1 - x)*(x + y*I))^(-(1)/(2)*alpha)* BesselJ(alpha, sqrt(2*(1 - x)*(x + y*I)))*((1)/(2)*(1 + x)*(x + y*I))^(-(1)/(2)*beta)* BesselI(beta, sqrt(2*(1 + x)*(x + y*I))) = sum((JacobiP(n, alpha, beta, x))/(GAMMA(n + alpha + 1)*GAMMA(n + beta + 1))*(x + y*I)^(n), n = 0..infinity)	(Divide[1,2]*(1 - x)*(x + y*I))^(-Divide[1,2]*\[Alpha])* BesselJ[\[Alpha], Sqrt[2*(1 - x)*(x + y*I)]]*(Divide[1,2]*(1 + x)*(x + y*I))^(-Divide[1,2]*\[Beta])* BesselI[\[Beta], Sqrt[2*(1 + x)*(x + y*I)]] == Sum[Divide[JacobiP[n, \[Alpha], \[Beta], x],Gamma[n + \[Alpha]+ 1]*Gamma[n + \[Beta]+ 1]]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [162 / 162] Result: Plus[Complex[0.981805922221423, -0.9438516537752855], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], Power[Gamma[Plus[2.5, n]], -2], JacobiP[n, 1.5, 1.5, 1.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5]} Result: Plus[Complex[1.6632758089192896, -2.584370418129778], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], Power[Gamma[Plus[1.5, n]], -1], Power[Gamma[Plus[2.5, n]], -1], JacobiP[n, 1.5, 0.5, 1.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 0.5]} ... skip entries to safe data
18.12.E3	${\displaystyle{\displaystyle(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n}}}$ (1+z)^{-\alpha-\beta-1}\*\genhyperF{2}{1}@@{\tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)}{\beta+1}{\frac{2(x+1)z}{(1+z)^{2}}} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{\alpha+\beta+1}{n}}{\Pochhammersym{\beta+1}{n}}\JacobipolyP{\alpha}{\beta}{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1 +(x + y*I))^(- alpha - beta - 1)* hypergeom([(1)/(2)*(alpha + beta + 1),(1)/(2)*(alpha + beta + 2)], [beta + 1], (2*(x + 1)*(x + y*I))/((1 +(x + y*I))^(2))) = sum((pochhammer(alpha + beta + 1, n))/(pochhammer(beta + 1, n))*JacobiP(n, alpha, beta, x)*(x + y*I)^(n), n = 0..infinity)	(1 +(x + y*I))^(- \[Alpha]- \[Beta]- 1)* HypergeometricPFQ[{Divide[1,2]*(\[Alpha]+ \[Beta]+ 1),Divide[1,2]*(\[Alpha]+ \[Beta]+ 2)}, {\[Beta]+ 1}, Divide[2*(x + 1)*(x + y*I),(1 +(x + y*I))^(2)]] == Sum[Divide[Pochhammer[\[Alpha]+ \[Beta]+ 1, n],Pochhammer[\[Beta]+ 1, n]]*JacobiP[n, \[Alpha], \[Beta], x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Manual Skip!	Failed [162 / 162] Result: Plus[Complex[0.08163265306122452, -5.551115123125783*^-17], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], JacobiP[n, 1.5, 1.5, 1.5], Power[Pochhammer[2.5, n], -1], Pochhammer[4.0, n]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 1.5]} Result: Plus[Complex[0.2040816326530612, -0.12244897959183688], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], JacobiP[n, 1.5, 0.5, 1.5], Power[Pochhammer[1.5, n], -1], Pochhammer[3.0, n]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5], Rule[β, 0.5]} ... skip entries to safe data
18.12.E4	${\displaystyle{\displaystyle(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{n}}}$ (1-2xz+z^{2})^{-\lambda} = \sum_{n=0}^{\infty}\ultrasphpoly{\lambda}{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1 - 2*x*(x + y*I)+(x + y*I)^(2))^(- lambda) = sum(GegenbauerC(n, lambda, x)*(x + y*I)^(n), n = 0..infinity)	(1 - 2*x*(x + y*I)+(x + y*I)^(2))^(- \[Lambda]) == Sum[GegenbauerC[n, \[Lambda], x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Manual Skip!	Successful [Tested: 180]
18.12.E4	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)z^{n}}}$ \sum_{n=0}^{\infty}\ultrasphpoly{\lambda}{n}@{x}z^{n} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{2\lambda}{n}}{\Pochhammersym{\lambda+\tfrac{1}{2}}{n}}\JacobipolyP{\lambda-\frac{1}{2}}{\lambda-\frac{1}{2}}{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	sum(GegenbauerC(n, lambda, x)*(x + y*I)^(n), n = 0..infinity) = sum((pochhammer(2*lambda, n))/(pochhammer(lambda +(1)/(2), n))*JacobiP(n, lambda -(1)/(2), lambda -(1)/(2), x)*(x + y*I)^(n), n = 0..infinity)	Sum[GegenbauerC[n, \[Lambda], x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None] == Sum[Divide[Pochhammer[2*\[Lambda], n],Pochhammer[\[Lambda]+Divide[1,2], n]]*JacobiP[n, \[Lambda]-Divide[1,2], \[Lambda]-Divide[1,2], x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Manual Skip!	Failed [162 / 180] Result: Plus[Complex[-1.5913916125772698, 0.33169349479585375], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], JacobiP[n, Plus[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[Rational[-1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], 1.5], Pochhammer[Times[2, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], n], Power[Pochhammer[Plus[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], n], -1]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[25.130585397727415, 13.271387895941402], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], JacobiP[n, Plus[Rational[-1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Plus[Rational[-1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], 1.5], Pochhammer[Times[2, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], n], Power[Pochhammer[Plus[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], n], -1]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
18.12.E5	${\displaystyle{\displaystyle\frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}}=\sum_{n=0}^{\infty}\frac{n+2\lambda}{2\lambda}C^{(\lambda)}_{n}\left(x\right)z^{n}}}$ \frac{1-xz}{(1-2xz+z^{2})^{\lambda+1}} = \sum_{n=0}^{\infty}\frac{n+2\lambda}{2\lambda}\ultrasphpoly{\lambda}{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1 - x*(x + y*I))/((1 - 2*x*(x + y*I)+(x + y*I)^(2))^(lambda + 1)) = sum((n + 2*lambda)/(2*lambda)*GegenbauerC(n, lambda, x)*(x + y*I)^(n), n = 0..infinity)	Divide[1 - x*(x + y*I),(1 - 2*x*(x + y*I)+(x + y*I)^(2))^(\[Lambda]+ 1)] == Sum[Divide[n + 2*\[Lambda],2*\[Lambda]]*GegenbauerC[n, \[Lambda], x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Manual Skip!	Skipped - Because timed out
18.12.E6	${\displaystyle{\displaystyle\Gamma\left(\lambda+\tfrac{1}{2}\right)e^{z\cos\theta}(\tfrac{1}{2}z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}\left(z\sin\theta\right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos\theta\right)}{{\left(2\lambda\right)_{n}}}z^{n}}}$ \EulerGamma@{\lambda+\tfrac{1}{2}}e^{z\cos@@{\theta}}(\tfrac{1}{2}z\sin@@{\theta})^{\frac{1}{2}-\lambda}\BesselJ{\lambda-\frac{1}{2}}@{z\sin@@{\theta}} = \sum_{n=0}^{\infty}\frac{\ultrasphpoly{\lambda}{n}@{\cos@@{\theta}}}{\Pochhammersym{2\lambda}{n}}z^{n}	${\displaystyle{\displaystyle 0\leq\theta,\theta\leq\pi,\Re((\lambda-\frac{1}{2})+k+1)>0,\Re(\lambda+\tfrac{1}{2})>0}}$	GAMMA(lambda +(1)/(2))*exp(z*cos(theta))*((1)/(2)*z*sin(theta))^((1)/(2)- lambda)* BesselJ(lambda -(1)/(2), z*sin(theta)) = sum((GegenbauerC(n, lambda, cos(theta)))/(pochhammer(2*lambda, n))*(z)^(n), n = 0..infinity)	Gamma[\[Lambda]+Divide[1,2]]*Exp[z*Cos[\[Theta]]]*(Divide[1,2]*z*Sin[\[Theta]])^(Divide[1,2]- \[Lambda])* BesselJ[\[Lambda]-Divide[1,2], z*Sin[\[Theta]]] == Sum[Divide[GegenbauerC[n, \[Lambda], Cos[\[Theta]]],Pochhammer[2*\[Lambda], n]]*(z)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Manual Skip!	Successful [Tested: 105]
18.12.E7	${\displaystyle{\displaystyle\frac{1-z^{2}}{1-2xz+z^{2}}=1+2\sum_{n=1}^{\infty}T_{n}\left(x\right)z^{n}}}$ \frac{1-z^{2}}{1-2xz+z^{2}} = 1+2\sum_{n=1}^{\infty}\ChebyshevpolyT{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1 -(x + y*I)^(2))/(1 - 2*x*(x + y*I)+(x + y*I)^(2)) = 1 + 2*sum(ChebyshevT(n, x)*(x + y*I)^(n), n = 1..infinity)	Divide[1 -(x + y*I)^(2),1 - 2*x*(x + y*I)+(x + y*I)^(2)] == 1 + 2*Sum[ChebyshevT[n, x]*(x + y*I)^(n), {n, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 18]
18.12.E8	${\displaystyle{\displaystyle\frac{1-xz}{1-2xz+z^{2}}=\sum_{n=0}^{\infty}T_{n}\left(x\right)z^{n}}}$ \frac{1-xz}{1-2xz+z^{2}} = \sum_{n=0}^{\infty}\ChebyshevpolyT{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1 - x*(x + y*I))/(1 - 2*x*(x + y*I)+(x + y*I)^(2)) = sum(ChebyshevT(n, x)*(x + y*I)^(n), n = 0..infinity)	Divide[1 - x*(x + y*I),1 - 2*x*(x + y*I)+(x + y*I)^(2)] == Sum[ChebyshevT[n, x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 18]
18.12.E9	${\displaystyle{\displaystyle-\ln\left(1-2xz+z^{2}\right)=2\sum_{n=1}^{\infty}\frac{T_{n}\left(x\right)}{n}z^{n}}}$ -\ln@{1-2xz+z^{2}} = 2\sum_{n=1}^{\infty}\frac{\ChebyshevpolyT{n}@{x}}{n}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	- ln(1 - 2*x*(x + y*I)+(x + y*I)^(2)) = 2*sum((ChebyshevT(n, x))/(n)*(x + y*I)^(n), n = 1..infinity)	- Log[1 - 2*x*(x + y*I)+(x + y*I)^(2)] == 2*Sum[Divide[ChebyshevT[n, x],n]*(x + y*I)^(n), {n, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [11 / 18] Result: 0.-6.283185308*I Test Values: {x = 3/2, y = 3/2} Result: .1e-9-6.283185308*I Test Values: {x = 3/2, y = 1/2} ... skip entries to safe data	Failed [8 / 18] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} Result: Complex[2.220446049250313*^-16, -6.283185307179586] Test Values: {Rule[x, 1.5], Rule[y, 0.5]} ... skip entries to safe data
18.12.E10	${\displaystyle{\displaystyle\frac{1}{1-2xz+z^{2}}=\sum_{n=0}^{\infty}U_{n}\left(x\right)z^{n}}}$ \frac{1}{1-2xz+z^{2}} = \sum_{n=0}^{\infty}\ChebyshevpolyU{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1)/(1 - 2*x*(x + y*I)+(x + y*I)^(2)) = sum(ChebyshevU(n, x)*(x + y*I)^(n), n = 0..infinity)	Divide[1,1 - 2*x*(x + y*I)+(x + y*I)^(2)] == Sum[ChebyshevU[n, x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 18]
18.12.E11	${\displaystyle{\displaystyle\frac{1}{\sqrt{1-2xz+z^{2}}}=\sum_{n=0}^{\infty}P_{n}\left(x\right)z^{n}}}$ \frac{1}{\sqrt{1-2xz+z^{2}}} = \sum_{n=0}^{\infty}\LegendrepolyP{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1)/(sqrt(1 - 2*x*(x + y*I)+(x + y*I)^(2))) = sum(LegendreP(n, x)*(x + y*I)^(n), n = 0..infinity)	Divide[1,Sqrt[1 - 2*x*(x + y*I)+(x + y*I)^(2)]] == Sum[LegendreP[n, x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [11 / 18] Result: -.7640216547e-17-1.069044968*I Test Values: {x = 3/2, y = 3/2} Result: -.1116612733e-18-1.632993162*I Test Values: {x = 3/2, y = 1/2} ... skip entries to safe data	Successful [Tested: 18]
18.12.E12	${\displaystyle{\displaystyle e^{xz}J_{0}\left(z\sqrt{1-x^{2}}\right)=\sum_{n=0}^{\infty}\frac{P_{n}\left(x\right)}{n!}z^{n}}}$ e^{xz}\BesselJ{0}@{z\sqrt{1-x^{2}}} = \sum_{n=0}^{\infty}\frac{\LegendrepolyP{n}@{x}}{n!}z^{n}	${\displaystyle{\displaystyle\Re(0+k+1)>0}}$	exp(x*(x + y*I))*BesselJ(0, (x + y*I)*sqrt(1 - (x)^(2))) = sum((LegendreP(n, x))/(factorial(n))*(x + y*I)^(n), n = 0..infinity)	Exp[x*(x + y*I)]*BesselJ[0, (x + y*I)*Sqrt[1 - (x)^(2)]] == Sum[Divide[LegendreP[n, x],(n)!]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 18]
18.12.E13	${\displaystyle{\displaystyle(1-z)^{-\alpha-1}\exp\left(\frac{xz}{z-1}\right)=\sum_{n=0}^{\infty}L^{(\alpha)}_{n}\left(x\right)z^{n}}}$ (1-z)^{-\alpha-1}\exp@{\frac{xz}{z-1}} = \sum_{n=0}^{\infty}\LaguerrepolyL[\alpha]{n}@{x}z^{n}	${\displaystyle{\displaystyle|z|<1}}$	(1 -(x + y*I))^(- alpha - 1)* exp((x*(x + y*I))/((x + y*I)- 1)) = sum(LaguerreL(n, alpha, x)*(x + y*I)^(n), n = 0..infinity)	(1 -(x + y*I))^(- \[Alpha]- 1)* Exp[Divide[x*(x + y*I),(x + y*I)- 1]] == Sum[LaguerreL[n, \[Alpha], x]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [54 / 54] Result: Plus[Complex[-1.4844951442502792, 1.2246448875280014], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], LaguerreL[n, 1.5, 1.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5]} Result: Plus[Complex[-1.0947197591668616, -2.83906516013942], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], LaguerreL[n, 0.5, 1.5]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 0.5]} ... skip entries to safe data
18.12.E14	${\displaystyle{\displaystyle\Gamma\left(\alpha+1\right)(xz)^{-\frac{1}{2}\alpha}e^{z}J_{\alpha}\left(2\sqrt{xz}\right)=\sum_{n=0}^{\infty}\frac{L^{(\alpha)}_{n}\left(x\right)}{{\left(\alpha+1\right)_{n}}}z^{n}}}$ \EulerGamma@{\alpha+1}(xz)^{-\frac{1}{2}\alpha}e^{z}\BesselJ{\alpha}@{2\sqrt{xz}} = \sum_{n=0}^{\infty}\frac{\LaguerrepolyL[\alpha]{n}@{x}}{\Pochhammersym{\alpha+1}{n}}z^{n}	${\displaystyle{\displaystyle\Re((\alpha)+k+1)>0,\Re(\alpha+1)>0}}$	GAMMA(alpha + 1)*(x*(x + y*I))^(-(1)/(2)*alpha)* exp(x + y*I)*BesselJ(alpha, 2*sqrt(x*(x + y*I))) = sum((LaguerreL(n, alpha, x))/(pochhammer(alpha + 1, n))*(x + y*I)^(n), n = 0..infinity)	Gamma[\[Alpha]+ 1]*(x*(x + y*I))^(-Divide[1,2]*\[Alpha])* Exp[x + y*I]*BesselJ[\[Alpha], 2*Sqrt[x*(x + y*I)]] == Sum[Divide[LaguerreL[n, \[Alpha], x],Pochhammer[\[Alpha]+ 1, n]]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [54 / 54] Result: Plus[Complex[1.918948179435534, -0.6639550064181744], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], LaguerreL[n, 1.5, 1.5], Power[Pochhammer[2.5, n], -1]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 1.5]} Result: Plus[Complex[1.8524178608069808, 1.376564839164941], Times[-1.0, NSum[Times[Power[Complex[1.5, -1.5], n], LaguerreL[n, 0.5, 1.5], Power[Pochhammer[1.5, n], -1]] Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[x, 1.5], Rule[y, -1.5], Rule[α, 0.5]} ... skip entries to safe data
18.12.E15	${\displaystyle{\displaystyle e^{2xz-z^{2}}=\sum_{n=0}^{\infty}\frac{H_{n}\left(x\right)}{n!}z^{n}}}$ e^{2xz-z^{2}} = \sum_{n=0}^{\infty}\frac{\HermitepolyH{n}@{x}}{n!}z^{n}		exp(2*x*(x + y*I)-(x + y*I)^(2)) = sum((HermiteH(n, x))/(factorial(n))*(x + y*I)^(n), n = 0..infinity)	Exp[2*x*(x + y*I)-(x + y*I)^(2)] == Sum[Divide[HermiteH[n, x],(n)!]*(x + y*I)^(n), {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 18]
18.14.E1	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq P^{(\alpha,\beta)}_{n}\left(1\right)}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x}| \leq \JacobipolyP{\alpha}{\beta}{n}@{1}	${\displaystyle{\displaystyle-1\leq x,x\leq 1,\alpha\geq\beta,\beta>-1}}$	abs(JacobiP(n, alpha, beta, x)) <= JacobiP(n, alpha, beta, 1)	Abs[JacobiP[n, \[Alpha], \[Beta], x]] <= JacobiP[n, \[Alpha], \[Beta], 1]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.14.E1	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}}}$ \JacobipolyP{\alpha}{\beta}{n}@{1} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}	${\displaystyle{\displaystyle-1\leq x,x\leq 1,\alpha\geq\beta,\beta>-1}}$	JacobiP(n, alpha, beta, 1) = (pochhammer(alpha + 1, n))/(factorial(n))	JacobiP[n, \[Alpha], \[Beta], 1] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 9]
18.14.E2	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq|P^{(\alpha,\beta)}_{n}\left(-1\right)|=\frac{{\left(\beta+1\right)_{n}}}{n!}}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x}| \leq |\JacobipolyP{\alpha}{\beta}{n}@{-1}|=\frac{\Pochhammersym{\beta+1}{n}}{n!}	${\displaystyle{\displaystyle-1\leq x,x\leq 1,\beta\geq\alpha,\alpha>-1}}$	abs(JacobiP(n, alpha, beta, x)) <= abs(JacobiP(n, alpha, beta, - 1)) = (pochhammer(beta + 1, n))/(factorial(n))	Abs[JacobiP[n, \[Alpha], \[Beta], x]] <= Abs[JacobiP[n, \[Alpha], \[Beta], - 1]] == Divide[Pochhammer[\[Beta]+ 1, n],(n)!]	Failure	Failure	Error	Failed [1 / 9] Result: False Test Values: {Rule[n, 1], Rule[x, 0.5], Rule[α, 2], Rule[β, Rational[1, 2]]}
18.14.E4	${\displaystyle{\displaystyle|C^{(\lambda)}_{n}\left(x\right)|\leq C^{(\lambda)}_{n}\left(1\right)}}$ |\ultrasphpoly{\lambda}{n}@{x}| \leq \ultrasphpoly{\lambda}{n}@{1}	${\displaystyle{\displaystyle-1\leq x,x\leq 1,\lambda>0}}$	abs(GegenbauerC(n, lambda, x)) <= GegenbauerC(n, lambda, 1)	Abs[GegenbauerC[n, \[Lambda], x]] <= GegenbauerC[n, \[Lambda], 1]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.14.E4	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(1\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}}}$ \ultrasphpoly{\lambda}{n}@{1} = \frac{\Pochhammersym{2\lambda}{n}}{n!}	${\displaystyle{\displaystyle-1\leq x,x\leq 1,\lambda>0}}$	GegenbauerC(n, lambda, 1) = (pochhammer(2*lambda, n))/(factorial(n))	GegenbauerC[n, \[Lambda], 1] == Divide[Pochhammer[2*\[Lambda], n],(n)!]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 9]
18.14.E5	${\displaystyle{\displaystyle|C^{(\lambda)}_{2m}\left(x\right)|\leq|C^{(\lambda)}_{2m}\left(0\right)|=\left|\frac{{\left(\lambda\right)_{m}}}{m!}\right|}}$ |\ultrasphpoly{\lambda}{2m}@{x}| \leq |\ultrasphpoly{\lambda}{2m}@{0}|=\left|\frac{\Pochhammersym{\lambda}{m}}{m!}\right|	${\displaystyle{\displaystyle-1\leq x,x\leq 1,-\tfrac{1}{2}<\lambda,\lambda<0}}$	abs(GegenbauerC(2*m, lambda, x)) <= abs(GegenbauerC(2*m, lambda, 0)) = abs((pochhammer(lambda, m))/(factorial(m)))	Abs[GegenbauerC[2*m, \[Lambda], x]] <= Abs[GegenbauerC[2*m, \[Lambda], 0]] == Abs[Divide[Pochhammer[\[Lambda], m],(m)!]]	Failure	Failure	Error	Skip - No test values generated
18.14.E6	${\displaystyle{\displaystyle|C^{(\lambda)}_{2m+1}\left(x\right)|<\frac{-2{\left(\lambda\right)_{m+1}}}{\left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m!}}}$ |\ultrasphpoly{\lambda}{2m+1}@{x}| < \frac{-2\Pochhammersym{\lambda}{m+1}}{\left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m!}	${\displaystyle{\displaystyle-1\leq x,x\leq 1,-\tfrac{1}{2}<\lambda,\lambda<0}}$	abs(GegenbauerC(2*m + 1, lambda, x)) < (- 2*pochhammer(lambda, m + 1))/(((2*m + 1)*(2*lambda + 2*m + 1))^((1)/(2))* factorial(m))	Abs[GegenbauerC[2*m + 1, \[Lambda], x]] < Divide[- 2*Pochhammer[\[Lambda], m + 1],((2*m + 1)*(2*\[Lambda]+ 2*m + 1))^(Divide[1,2])* (m)!]	Failure	Failure	Error	Skip - No test values generated
18.14.E7	${\displaystyle{\displaystyle(n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}\lambda}|C^{(\lambda)}_{n}\left(x\right)|<\frac{2^{1-\lambda}}{\Gamma\left(\lambda\right)}}}$ (n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}\lambda}|\ultrasphpoly{\lambda}{n}@{x}| < \frac{2^{1-\lambda}}{\EulerGamma@{\lambda}}	${\displaystyle{\displaystyle-1\leq x,x\leq 1,0<\lambda,\lambda<1,\Re(\lambda)>0}}$	(n + lambda)^(1 - lambda)*(1 - (x)^(2))^((1)/(2)*lambda)*abs(GegenbauerC(n, lambda, x)) < ((2)^(1 - lambda))/(GAMMA(lambda))	(n + \[Lambda])^(1 - \[Lambda])*(1 - (x)^(2))^(Divide[1,2]*\[Lambda])*Abs[GegenbauerC[n, \[Lambda], x]] < Divide[(2)^(1 - \[Lambda]),Gamma[\[Lambda]]]	Skipped - Unable to analyze test case: Null	Skipped - Unable to analyze test case: Null	-	-
18.14.E8	${\displaystyle{\displaystyle e^{-\frac{1}{2}x}\left|L^{(\alpha)}_{n}\left(x\right)\right|\leq L^{(\alpha)}_{n}\left(0\right)}}$ e^{-\frac{1}{2}x}\left|\LaguerrepolyL[\alpha]{n}@{x}\right| \leq \LaguerrepolyL[\alpha]{n}@{0}	${\displaystyle{\displaystyle 0\leq x,x<\infty,\alpha\geq 0}}$	exp(-(1)/(2)*x)*abs(LaguerreL(n, alpha, x)) <= LaguerreL(n, alpha, 0)	Exp[-Divide[1,2]*x]*Abs[LaguerreL[n, \[Alpha], x]] <= LaguerreL[n, \[Alpha], 0]	Missing Macro Error	Failure	-	Successful [Tested: 27]
18.14.E8	${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(0\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}}}$ \LaguerrepolyL[\alpha]{n}@{0} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}	${\displaystyle{\displaystyle 0\leq x,x<\infty,\alpha\geq 0}}$	LaguerreL(n, alpha, 0) = (pochhammer(alpha + 1, n))/(factorial(n))	LaguerreL[n, \[Alpha], 0] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]	Missing Macro Error	Successful	-	Successful [Tested: 9]
18.14.E9	${\displaystyle{\displaystyle\frac{1}{(2^{n}n!)^{\frac{1}{2}}}e^{-\frac{1}{2}x^{2}}|H_{n}\left(x\right)|\leq 1}}$ \frac{1}{(2^{n}n!)^{\frac{1}{2}}}e^{-\frac{1}{2}x^{2}}|\HermitepolyH{n}@{x}| \leq 1	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	(1)/(((2)^(n)* factorial(n))^((1)/(2)))*exp(-(1)/(2)*(x)^(2))*abs(HermiteH(n, x)) <= 1	Divide[1,((2)^(n)* (n)!)^(Divide[1,2])]*Exp[-Divide[1,2]*(x)^(2)]*Abs[HermiteH[n, x]] <= 1	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.14.E10	${\displaystyle{\displaystyle(P_{n}\left(x\right))^{2}\geq P_{n-1}\left(x\right)P_{n+1}\left(x\right)}}$ (\LegendrepolyP{n}@{x})^{2} \geq \LegendrepolyP{n-1}@{x}\LegendrepolyP{n+1}@{x}	${\displaystyle{\displaystyle-1\leq x,x\leq 1}}$	(LegendreP(n, x))^(2) >= LegendreP(n - 1, x)*LegendreP(n + 1, x)	(LegendreP[n, x])^(2) >= LegendreP[n - 1, x]*LegendreP[n + 1, x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
18.14.E11	${\displaystyle{\displaystyle(R_{n}(x))^{2}\geq R_{n-1}(x)R_{n+1}(x)}}$ (R_{n}(x))^{2} \geq R_{n-1}(x)R_{n+1}(x)	${\displaystyle{\displaystyle-1\leq x,x\leq 1,\beta\geq\alpha,\alpha>-1}}$	(R[n](x))^(2) >= R[n - 1](x)* R[n + 1](x)	(Subscript[R, n][x])^(2) >= Subscript[R, n - 1][x]* Subscript[R, n + 1][x]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.14.E12	${\displaystyle{\displaystyle(L^{(\alpha)}_{n}\left(x\right))^{2}\geq L^{(\alpha)}_{n-1}\left(x\right)L^{(\alpha)}_{n+1}\left(x\right)}}$ (\LaguerrepolyL[\alpha]{n}@{x})^{2} \geq \LaguerrepolyL[\alpha]{n-1}@{x}\LaguerrepolyL[\alpha]{n+1}@{x}	${\displaystyle{\displaystyle 0\leq x,x<\infty,\alpha\geq 0}}$	(LaguerreL(n, alpha, x))^(2) >= LaguerreL(n - 1, alpha, x)*LaguerreL(n + 1, alpha, x)	(LaguerreL[n, \[Alpha], x])^(2) >= LaguerreL[n - 1, \[Alpha], x]*LaguerreL[n + 1, \[Alpha], x]	Missing Macro Error	Failure	-	Successful [Tested: 27]
18.14.E13	${\displaystyle{\displaystyle(H_{n}\left(x\right))^{2}\geq H_{n-1}\left(x\right)H_{n+1}\left(x\right)}}$ (\HermitepolyH{n}@{x})^{2} \geq \HermitepolyH{n-1}@{x}\HermitepolyH{n+1}@{x}	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	(HermiteH(n, x))^(2) >= HermiteH(n - 1, x)*HermiteH(n + 1, x)	(HermiteH[n, x])^(2) >= HermiteH[n - 1, x]*HermiteH[n + 1, x]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
18.14.E14	${\displaystyle{\displaystyle-1=x_{n,0}}}$ -1 = x_{n,0}		- 1 = x[n , 0]	- 1 == Subscript[x, n , 0]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.14.E15	${\displaystyle{\displaystyle x_{n,m}\leq(\beta-\alpha)/(\alpha+\beta+1)\leq x_{n,m+1}}}$ x_{n,m} \leq (\beta-\alpha)/(\alpha+\beta+1)\leq x_{n,m+1}		x[n , m] <= (beta - alpha)/(alpha + beta + 1) <= x[n , m + 1]	Subscript[x, n , m] <= (\[Beta]- \[Alpha])/(\[Alpha]+ \[Beta]+ 1) <= Subscript[x, n , m + 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.14#Ex1	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|>|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|		abs(JacobiP(n, alpha, beta, x[n , 0])) > abs(JacobiP(n, alpha, beta, x[n , 1]))	Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] > Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]	Failure	Failure	Failed [184 / 300] Result: 2.500000000 < 2.500000000 Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 1} Result: 4.871793818 < 4.871793818 Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [184 / 300] Result: False Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: False Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.14#Ex2	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|>|P^{(\alpha,\beta)}_{n}\left(x_{n,n-1}\right)|}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-1}}|	${\displaystyle{\displaystyle\alpha>-\tfrac{1}{2},\beta>-\tfrac{1}{2}.}}$	abs(JacobiP(n, alpha, beta, x[n , n])) > abs(JacobiP(n, alpha, beta, x[n , n - 1]))	Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]] > Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]]	Error	Failure	-	Skip - No test values generated
18.14#Ex3	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|<|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|		abs(JacobiP(n, alpha, beta, x[n , 0])) < abs(JacobiP(n, alpha, beta, x[n , 1]))	Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] < Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]	Failure	Failure	Failed [184 / 300] Result: 2.500000000 < 2.500000000 Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 1} Result: 4.871793820 < 4.871793820 Test Values: {alpha = 3/2, beta = 3/2, x[n,0] = 1/2*3^(1/2)+1/2*I, x[n,1] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [184 / 300] Result: False Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: False Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.14#Ex4	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|<|P^{(\alpha,\beta)}_{n}\left(x_{n,n-1}\right)|}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-1}}|	${\displaystyle{\displaystyle-1<\alpha,\alpha<-\tfrac{1}{2},-1<\beta,\beta<-\tfrac{1}{2}.}}$	abs(JacobiP(n, alpha, beta, x[n , n])) < abs(JacobiP(n, alpha, beta, x[n , n - 1]))	Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n]]] < Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - 1]]]	Error	Failure	-	Skip - No test values generated
18.14.E18	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|<|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| < |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|	${\displaystyle{\displaystyle-1<\beta,\beta\leq-\tfrac{1}{2}}}$	abs(JacobiP(n, alpha, beta, x[n , 0])) < abs(JacobiP(n, alpha, beta, x[n , 1]))	Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] < Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]	Failure	Failure	Error	Skip - No test values generated
18.14.E19	${\displaystyle{\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|>|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|}}$ |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,0}}| > |\JacobipolyP{\alpha}{\beta}{n}@{x_{n,1}}|	${\displaystyle{\displaystyle-1<\alpha,\alpha\leq-\tfrac{1}{2}}}$	abs(JacobiP(n, alpha, beta, x[n , 0])) > abs(JacobiP(n, alpha, beta, x[n , 1]))	Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 0]]] > Abs[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , 1]]]	Failure	Failure	Error	Skip - No test values generated
18.14.E20	${\displaystyle{\displaystyle\left|\frac{P^{(\alpha,\beta)}_{n}\left(x_{n,n-m}\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}\right|>\left|\frac{P^{(\alpha,\beta)}_{n+1}\left(x_{n+1,n-m+1}\right)}{P^{(\alpha,\beta)}_{n+1}\left(1\right)}\right|}}$ \left|\frac{\JacobipolyP{\alpha}{\beta}{n}@{x_{n,n-m}}}{\JacobipolyP{\alpha}{\beta}{n}@{1}}\right| > \left|\frac{\JacobipolyP{\alpha}{\beta}{n+1}@{x_{n+1,n-m+1}}}{\JacobipolyP{\alpha}{\beta}{n+1}@{1}}\right|	${\displaystyle{\displaystyle\alpha=\beta,\beta>-\tfrac{1}{2},m=1}}$	abs((JacobiP(n, alpha, beta, x[n , n - m]))/(JacobiP(n, alpha, beta, 1))) > abs((JacobiP(n + 1, alpha, beta, x[n + 1 , n - m + 1]))/(JacobiP(n + 1, alpha, beta, 1)))	Abs[Divide[JacobiP[n, \[Alpha], \[Beta], Subscript[x, n , n - m]],JacobiP[n, \[Alpha], \[Beta], 1]]] > Abs[Divide[JacobiP[n + 1, \[Alpha], \[Beta], Subscript[x, n + 1 , n - m + 1]],JacobiP[n + 1, \[Alpha], \[Beta], 1]]]	Failure	Failure	Failed [188 / 300] Result: 1.113552873 < 1.000000000 Test Values: {alpha = 3/2, beta = 3/2, x[n,n-m] = 1/2*3^(1/2)+1/2*I, x[n+1,n-m+1] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1} Result: 1.400000001 < 1.113552873 Test Values: {alpha = 3/2, beta = 3/2, x[n,n-m] = 1/2*3^(1/2)+1/2*I, x[n+1,n-m+1] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2} ... skip entries to safe data	Failed [234 / 300] Result: False Test Values: {Rule[m, 1], Rule[n, 1], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, n], Plus[1, Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: False Test Values: {Rule[m, 1], Rule[n, 2], Rule[α, 1.5], Rule[β, 1.5], Rule[Subscript[x, n, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, n], Plus[1, Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.14.E21	${\displaystyle{\displaystyle 0=x_{n,0}}}$ 0 = x_{n,0}		0 = x[n , 0]	0 == Subscript[x, n , 0]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.14.E22	${\displaystyle{\displaystyle x_{n,m}\leq\alpha+\tfrac{1}{2}}}$ x_{n,m} \leq \alpha+\tfrac{1}{2}		x[n , m] <= alpha +(1)/(2)	Subscript[x, n , m] <= \[Alpha]+Divide[1,2]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.14#Ex5	${\displaystyle{\displaystyle|L^{(\alpha)}_{n}\left(x_{n,0}\right)|>|L^{(\alpha)}_{n}\left(x_{n,1}\right)|}}$ |\LaguerrepolyL[\alpha]{n}@{x_{n,0}}| > |\LaguerrepolyL[\alpha]{n}@{x_{n,1}}|		abs(LaguerreL(n, alpha, x[n , 0])) > abs(LaguerreL(n, alpha, x[n , 1]))	Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 0]]] > Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 1]]]	Missing Macro Error	Failure	-	Failed [165 / 300] Result: False Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: False Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.14#Ex6	${\displaystyle{\displaystyle|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|>|L^{(\alpha)}_{n}\left(x_{n,n-2}\right)|}}$ |\LaguerrepolyL[\alpha]{n}@{x_{n,n-1}}| > |\LaguerrepolyL[\alpha]{n}@{x_{n,n-2}}|		abs(LaguerreL(n, alpha, x[n , n - 1])) > abs(LaguerreL(n, alpha, x[n , n - 2]))	Abs[LaguerreL[n, \[Alpha], Subscript[x, n , n - 1]]] > Abs[LaguerreL[n, \[Alpha], Subscript[x, n , n - 2]]]	Missing Macro Error	Failure	-	Failed [165 / 300] Result: False Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[Subscript[x, n, Plus[-2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: False Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[Subscript[x, n, Plus[-2, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.14.E24	${\displaystyle{\displaystyle|L^{(\alpha)}_{n}\left(x_{n,0}\right)|<|L^{(\alpha)}_{n}\left(x_{n,1}\right)|}}$ |\LaguerrepolyL[\alpha]{n}@{x_{n,0}}| < |\LaguerrepolyL[\alpha]{n}@{x_{n,1}}|		abs(LaguerreL(n, alpha, x[n , 0])) < abs(LaguerreL(n, alpha, x[n , 1]))	Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 0]]] < Abs[LaguerreL[n, \[Alpha], Subscript[x, n , 1]]]	Missing Macro Error	Failure	-	Failed [165 / 300] Result: False Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: False Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[Subscript[x, n, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.15.E6	${\displaystyle{\displaystyle(\sin\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\cos\tfrac{1}{2}\theta)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)=\frac{\Gamma\left(n+\alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*\left(\theta^{\frac{1}{2}}J_{\alpha}\left(\rho\theta\right)\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}J_{\alpha+1}\left(\rho\theta\right)\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right)}}$ (\sin@@{\tfrac{1}{2}\theta})^{\alpha+\frac{1}{2}}(\cos@@{\tfrac{1}{2}\theta})^{\beta+\frac{1}{2}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\theta}} = \frac{\EulerGamma@{n+\alpha+1}}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*\left(\theta^{\frac{1}{2}}\BesselJ{\alpha}@{\rho\theta}\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}\BesselJ{\alpha+1}@{\rho\theta}\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right)	${\displaystyle{\displaystyle\Re((\alpha)+k+1)>0,\Re((\alpha+1)+k+1)>0,\Re(n+\alpha+1)>0}}$	(sin((1)/(2)*theta))^(alpha +(1)/(2))*(cos((1)/(2)*theta))^(beta +(1)/(2))* JacobiP(n, alpha, beta, cos(theta)) = (GAMMA(n + alpha + 1))/((2)^((1)/(2))*(n +(1)/(2)*(alpha + beta + 1))^(alpha)* factorial(n))*((theta)^((1)/(2))* BesselJ(alpha, (n +(1)/(2)*(alpha + beta + 1))*theta)*sum((A[m](theta))/((n +(1)/(2)*(alpha + beta + 1))^(2*m)), m = 0..M)+ (theta)^((3)/(2))* BesselJ(alpha + 1, (n +(1)/(2)*(alpha + beta + 1))*theta)*sum((B[m](theta))/((n +(1)/(2)*(alpha + beta + 1))^(2*m + 1)), m = 0..M - 1)+ varepsilon[M]((n +(1)/(2)*(alpha + beta + 1)), theta))	(Sin[Divide[1,2]*\[Theta]])^(\[Alpha]+Divide[1,2])*(Cos[Divide[1,2]*\[Theta]])^(\[Beta]+Divide[1,2])* JacobiP[n, \[Alpha], \[Beta], Cos[\[Theta]]] == Divide[Gamma[n + \[Alpha]+ 1],(2)^(Divide[1,2])*(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^\[Alpha]* (n)!]*(\[Theta]^(Divide[1,2])* BesselJ[\[Alpha], (n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))*\[Theta]]*Sum[Divide[Subscript[A, m][\[Theta]],(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2*m)], {m, 0, M}, GenerateConditions->None]+ \[Theta]^(Divide[3,2])* BesselJ[\[Alpha]+ 1, (n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))*\[Theta]]*Sum[Divide[Subscript[B, m][\[Theta]],(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2*m + 1)], {m, 0, M - 1}, GenerateConditions->None]+ Subscript[\[CurlyEpsilon], M][(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1)), \[Theta]])	Failure	Failure	Skipped - Because timed out	Skipped - Because timed out
18.15.E24	${\displaystyle{\displaystyle\mu=2n+1}}$ \mu = 2n+1		mu = 2*n + 1	\[Mu] == 2*n + 1	Skipped - no semantic math	Skipped - no semantic math	-	-
18.15.E28	${\displaystyle{\displaystyle H_{n}\left(x\right)=2^{\frac{1}{4}(\mu^{2}-1)}e^{\frac{1}{2}\mu^{2}t^{2}}U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)}}$ \HermitepolyH{n}@{x} = 2^{\frac{1}{4}(\mu^{2}-1)}e^{\frac{1}{2}\mu^{2}t^{2}}\paraU@{-\tfrac{1}{2}\mu^{2}}{\mu t\sqrt{2}}		HermiteH(n, x) = (2)^((1)/(4)*((mu)^(2)- 1))* exp((1)/(2)*(mu)^(2)* (t)^(2))*CylinderU(-(1)/(2)*(mu)^(2), mu*t*sqrt(2))	HermiteH[n, x] == (2)^(Divide[1,4]*(\[Mu]^(2)- 1))* Exp[Divide[1,2]*\[Mu]^(2)* (t)^(2)]*ParabolicCylinderD[- 1/2 -(-Divide[1,2]*\[Mu]^(2)), \[Mu]*t*Sqrt[2]]	Failure	Failure	Failed [300 / 300] Result: -1.440969060-2.714107233*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, n = 1} Result: 2.559030940-2.714107233*I Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, n = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-1.440969055661161, -2.714107231302052] Test Values: {Rule[n, 1], Rule[t, -1.5], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[2.559030944338839, -2.714107231302052] Test Values: {Rule[n, 2], Rule[t, -1.5], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.16.E1	${\displaystyle{\displaystyle 0<\theta_{n,1}}}$ 0 < \theta_{n,1}		0 < theta[n , 1]	0 < Subscript[\[Theta], n , 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E2	${\displaystyle{\displaystyle\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{1}{2}}\leq\theta_{n,m}}}$ \frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{1}{2}} \leq \theta_{n,m}		((m -(1)/(2))*Pi)/(n +(1)/(2)) <= theta[n , m]	Divide[(m -Divide[1,2])*Pi,n +Divide[1,2]] <= Subscript[\[Theta], n , m]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E3	${\displaystyle{\displaystyle\frac{(m-\tfrac{1}{2})\pi}{n}\leq\theta_{n,m}}}$ \frac{(m-\tfrac{1}{2})\pi}{n} \leq \theta_{n,m}	${\displaystyle{\displaystyle\alpha=\beta}}$	((m -(1)/(2))*Pi)/(n) <= theta[n , m]	Divide[(m -Divide[1,2])*Pi,n] <= Subscript[\[Theta], n , m]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E4	${\displaystyle{\displaystyle\frac{\left(m+\tfrac{1}{2}(\alpha+\beta-1)\right)\pi}{\rho}<\theta_{n,m}}}$ \frac{\left(m+\tfrac{1}{2}(\alpha+\beta-1)\right)\pi}{\rho} < \theta_{n,m}		((m +(1)/(2)*(alpha + beta - 1))*Pi)/(n +(1)/(2)*(alpha + beta + 1)) < theta[n , m]	Divide[(m +Divide[1,2]*(\[Alpha]+ \[Beta]- 1))*Pi,n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1)] < Subscript[\[Theta], n , m]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E5	${\displaystyle{\displaystyle\theta_{n,m}>\frac{\left(m+\tfrac{1}{2}\alpha-\tfrac{1}{4}\right){\pi}}{n+\alpha+\tfrac{1}{2}}}}$ \theta_{n,m} > \frac{\left(m+\tfrac{1}{2}\alpha-\tfrac{1}{4}\right){\pi}}{n+\alpha+\tfrac{1}{2}}	${\displaystyle{\displaystyle\alpha=\beta}}$	theta[n , m] > ((m +(1)/(2)*alpha -(1)/(4))*Pi)/(n + alpha +(1)/(2))	Subscript[\[Theta], n , m] > Divide[(m +Divide[1,2]*\[Alpha]-Divide[1,4])*Pi,n + \[Alpha]+Divide[1,2]]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E6	${\displaystyle{\displaystyle\theta_{n,m}\leq\frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{12}\left(1-\alpha^{2}-3\beta^{2}\right)\right)^{\frac{1}{2}}}}}$ \theta_{n,m} \leq \frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{12}\left(1-\alpha^{2}-3\beta^{2}\right)\right)^{\frac{1}{2}}}		theta[n , m] <= (j[alpha , m])/(((n +(1)/(2)*(alpha + beta + 1))^(2)+(1)/(12)*(1 - (alpha)^(2)- 3*(beta)^(2)))^((1)/(2)))	Subscript[\[Theta], n , m] <= Divide[Subscript[j, \[Alpha], m],((n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2)+Divide[1,12]*(1 - \[Alpha]^(2)- 3*\[Beta]^(2)))^(Divide[1,2])]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E7	${\displaystyle{\displaystyle\theta_{n,m}\geq\frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{4}-\tfrac{1}{2}(\alpha^{2}+\beta^{2})-\pi^{-2}(1-4\alpha^{2})\right)^{\frac{1}{2}}}}}$ \theta_{n,m} \geq \frac{j_{\alpha,m}}{\left(\rho^{2}+\tfrac{1}{4}-\tfrac{1}{2}(\alpha^{2}+\beta^{2})-\pi^{-2}(1-4\alpha^{2})\right)^{\frac{1}{2}}}		theta[n , m] >= (j[alpha , m])/(((n +(1)/(2)*(alpha + beta + 1))^(2)+(1)/(4)-(1)/(2)*((alpha)^(2)+ (beta)^(2))- (Pi)^(- 2)*(1 - 4*(alpha)^(2)))^((1)/(2)))	Subscript[\[Theta], n , m] >= Divide[Subscript[j, \[Alpha], m],((n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2)+Divide[1,4]-Divide[1,2]*(\[Alpha]^(2)+ \[Beta]^(2))- (Pi)^(- 2)*(1 - 4*\[Alpha]^(2)))^(Divide[1,2])]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E9	${\displaystyle{\displaystyle 0<x_{n,1}}}$ 0 < x_{n,1}		0 < x[n , 1]	0 < Subscript[x, n , 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E10	${\displaystyle{\displaystyle x_{n,m}>\ifrac{j_{\alpha,m}^{2}}{\nu}}}$ x_{n,m} > \ifrac{j_{\alpha,m}^{2}}{\nu}		x[n , m] > ((j[alpha , m])^(2))/(4*n + 2*alpha + 2)	Subscript[x, n , m] > Divide[(Subscript[j, \[Alpha], m])^(2),4*n + 2*\[Alpha]+ 2]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E11	${\displaystyle{\displaystyle x_{n,m}<(4m+2\alpha+2)\left(2m+\alpha+1+\left((2m+\alpha+1)^{2}+\tfrac{1}{4}-\alpha^{2}\right)^{\frac{1}{2}}\right)\Big{/}\nu}}$ x_{n,m} < (4m+2\alpha+2)\left(2m+\alpha+1+\left((2m+\alpha+1)^{2}+\tfrac{1}{4}-\alpha^{2}\right)^{\frac{1}{2}}\right)\Big{/}\nu		x[n , m] < (4*m + 2*alpha + 2)*(2*m + alpha + 1 +((2*m + alpha + 1)^(2)+(1)/(4)- (alpha)^(2))^((1)/(2)))/(4*n + 2*alpha + 2)	Subscript[x, n , m] < (4*m + 2*\[Alpha]+ 2)*(2*m + \[Alpha]+ 1 +((2*m + \[Alpha]+ 1)^(2)+Divide[1,4]- \[Alpha]^(2))^(Divide[1,2]))/(4*n + 2*\[Alpha]+ 2)	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E12	${\displaystyle{\displaystyle x_{n,1}\geq\frac{2n^{2}+\alpha n-n+2\alpha+2-2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}}}$ x_{n,1} \geq \frac{2n^{2}+\alpha n-n+2\alpha+2-2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}		x[n , 1] >= (2*(n)^(2)+ alpha*n - n + 2*alpha + 2 - 2*(n - 1)*sqrt((n)^(2)+(n + 2)*(alpha + 1)))/(n + 2)	Subscript[x, n , 1] >= Divide[2*(n)^(2)+ \[Alpha]*n - n + 2*\[Alpha]+ 2 - 2*(n - 1)*Sqrt[(n)^(2)+(n + 2)*(\[Alpha]+ 1)],n + 2]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E13	${\displaystyle{\displaystyle x_{n,n}\leq\frac{2n^{2}+\alpha n-n+2\alpha+2+2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}}}$ x_{n,n} \leq \frac{2n^{2}+\alpha n-n+2\alpha+2+2(n-1)\sqrt{n^{2}+(n+2)(\alpha+1)}}{n+2}		x[n , n] <= (2*(n)^(2)+ alpha*n - n + 2*alpha + 2 + 2*(n - 1)*sqrt((n)^(2)+(n + 2)*(alpha + 1)))/(n + 2)	Subscript[x, n , n] <= Divide[2*(n)^(2)+ \[Alpha]*n - n + 2*\[Alpha]+ 2 + 2*(n - 1)*Sqrt[(n)^(2)+(n + 2)*(\[Alpha]+ 1)],n + 2]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.16.E16	${\displaystyle{\displaystyle(2n+1)^{\frac{1}{2}}>x_{n,1}}}$ (2n+1)^{\frac{1}{2}} > x_{n,1}		(2*n + 1)^((1)/(2)) > x[n , 1]	(2*n + 1)^(Divide[1,2]) > Subscript[x, n , 1]	Failure	Failure	Failed [1 / 30] Result: 2. < 1.732050808 Test Values: {x[n,1] = 2, n = 1}	Failed [13 / 30] Result: Greater[1.7320508075688772, Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[n, 1], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Greater[2.23606797749979, Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[n, 2], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.16.E16	${\displaystyle{\displaystyle x_{n,1}>x_{n,2}}}$ x_{n,1} > x_{n,2}		x[n , 1] > x[n , 2]	Subscript[x, n , 1] > Subscript[x, n , 2]	Failure	Failure	Failed [75 / 300] Result: .8660254040+.5000000000*I < .8660254040+.5000000000*I Test Values: {x[n,1] = 1/2*3^(1/2)+1/2*I, x[n,2] = 1/2*3^(1/2)+1/2*I, n = 1} Result: .8660254040+.5000000000*I < .8660254040+.5000000000*I Test Values: {x[n,1] = 1/2*3^(1/2)+1/2*I, x[n,2] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [255 / 300] Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[n, 1], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[n, 2], Rule[Subscript[x, n, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, n, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.16.E16	${\displaystyle{\displaystyle x_{n,\left\lfloor n/2\right\rfloor}>0}}$ x_{n,\floor{n/2}} > 0		x[n , floor(n/2)] > 0	Subscript[x, n , Floor[n/2]] > 0	Failure	Failure	Failed [9 / 30] Result: 0. < -1.500000000 Test Values: {x[n,floor(1/2*n)] = -3/2, n = 1} Result: 0. < -1.500000000 Test Values: {x[n,floor(1/2*n)] = -3/2, n = 2} ... skip entries to safe data	Failed [21 / 30] Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0] Test Values: {Rule[n, 1], Rule[Subscript[x, n, Floor[Times[Rational[1, 2], n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0] Test Values: {Rule[n, 2], Rule[Subscript[x, n, Floor[Times[Rational[1, 2], n]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data