18.33

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
18.33.E2	${\displaystyle{\displaystyle\phi_{n}(z)=\kappa_{n}z^{n}+\sum_{\ell=1}^{n}\kappa_{n,n-\ell}z^{n-\ell}}}$ \phi_{n}(z) = \kappa_{n}z^{n}+\sum_{\ell=1}^{n}\kappa_{n,n-\ell}z^{n-\ell}		phi[n](z) = kappa[n]*(z)^(n)+ sum(kappa[n , n - ell]*(z)^(n - ell), ell = 1..n)	Subscript[\[Phi], n][z] == Subscript[\[Kappa], n]*(z)^(n)+ Sum[Subscript[\[Kappa], n , n - \[ScriptL]]*(z)^(n - \[ScriptL]), {\[ScriptL], 1, n}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33.E3	${\displaystyle{\displaystyle\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline{z}^{-1}})}}}$ \phi_{n}^{*}(z) = z^{n}\conj{\phi_{n}(\conj{z}^{-1})}		(phi[n])^(*)(z) = (z)^(n)* conjugate(phi[n]((conjugate(z))^(- 1)))	(Subscript[\[Phi], n])^(*)[z] == (z)^(n)* Conjugate[Subscript[\[Phi], n][(Conjugate[z])^(- 1)]]	Error	Failure	Skip - symbolical successful subtest	Error
18.33.E3	${\displaystyle{\displaystyle z^{n}\overline{\phi_{n}({\overline{z}^{-1}})}={\kappa_{n}}+\sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell}z^{\ell}}}$ z^{n}\conj{\phi_{n}(\conj{z}^{-1})} = {\kappa_{n}}+\sum_{\ell=1}^{n}\conj{\kappa}_{n,n-\ell}z^{\ell}		(z)^(n)* conjugate(phi[n]((conjugate(z))^(- 1))) = kappa[n]+ sum(conjugate(kappa)[n , n - ell]*(z)^(ell), ell = 1..n)	(z)^(n)* Conjugate[Subscript[\[Phi], n][(Conjugate[z])^(- 1)]] == Subscript[\[Kappa], n]+ Sum[Subscript[Conjugate[\[Kappa]], n , n - \[ScriptL]]*(z)^\[ScriptL], {\[ScriptL], 1, n}, GenerateConditions->None]	Aborted	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.0, -0.9999999999999999], Times[Complex[-0.8660254037844387, -0.49999999999999994], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 1, 0]]] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[κ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.1339745962155613, -0.49999999999999994], Times[Complex[-0.5000000000000001, -0.8660254037844386], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 2, 0]], Times[Complex[-0.8660254037844387, -0.49999999999999994], Subscript[Complex[0.8660254037844387, -0.49999999999999994], 2, 1]]] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[κ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.33.E4	${\displaystyle{\displaystyle\kappa_{n}z\phi_{n}(z)=\kappa_{n+1}\phi_{n+1}(z)-\phi_{n+1}(0)\phi_{n+1}^{*}(z)}}$ \kappa_{n}z\phi_{n}(z) = \kappa_{n+1}\phi_{n+1}(z)-\phi_{n+1}(0)\phi_{n+1}^{*}(z)		kappa[n]*z*phi[n](z) = kappa[n + 1]*phi[n + 1](z)- phi[n + 1](0)* (phi[n + 1])^(*)(z)	Subscript[\[Kappa], n]*z*Subscript[\[Phi], n][z] == Subscript[\[Kappa], n + 1]*Subscript[\[Phi], n + 1][z]- Subscript[\[Phi], n + 1][0]* (Subscript[\[Phi], n + 1])^(*)[z]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33.E5	${\displaystyle{\displaystyle\kappa_{n}\phi_{n+1}(z)=\kappa_{n+1}z\phi_{n}(z)+\phi_{n+1}(0)\phi_{n}^{*}(z)}}$ \kappa_{n}\phi_{n+1}(z) = \kappa_{n+1}z\phi_{n}(z)+\phi_{n+1}(0)\phi_{n}^{*}(z)		kappa[n]*phi[n + 1](z) = kappa[n + 1]*z*phi[n](z)+ phi[n + 1](0)* (phi[n])^(*)(z)	Subscript[\[Kappa], n]*Subscript[\[Phi], n + 1][z] == Subscript[\[Kappa], n + 1]*z*Subscript[\[Phi], n][z]+ Subscript[\[Phi], n + 1][0]* (Subscript[\[Phi], n])^(*)[z]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33.E6	${\displaystyle{\displaystyle\kappa_{n}\phi_{n}(0)\phi_{n+1}(z)+\kappa_{n-1}\phi_{n+1}(0)z\phi_{n-1}(z)=\left(\kappa_{n}\phi_{n+1}(0)+\kappa_{n+1}\phi_{n}(0)z\right)\phi_{n}(z)}}$ \kappa_{n}\phi_{n}(0)\phi_{n+1}(z)+\kappa_{n-1}\phi_{n+1}(0)z\phi_{n-1}(z) = \left(\kappa_{n}\phi_{n+1}(0)+\kappa_{n+1}\phi_{n}(0)z\right)\phi_{n}(z)		kappa[n]*phi[n](0)* phi[n + 1](z)+ kappa[n - 1]*phi[n + 1](0)* z*phi[n - 1](z) = (kappa[n]*phi[n + 1](0)+ kappa[n + 1]*phi[n](0)* z)*phi[n](z)	Subscript[\[Kappa], n]*Subscript[\[Phi], n][0]* Subscript[\[Phi], n + 1][z]+ Subscript[\[Kappa], n - 1]*Subscript[\[Phi], n + 1][0]* z*Subscript[\[Phi], n - 1][z] == (Subscript[\[Kappa], n]*Subscript[\[Phi], n + 1][0]+ Subscript[\[Kappa], n + 1]*Subscript[\[Phi], n][0]* z)*Subscript[\[Phi], n][z]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33#Ex1	${\displaystyle{\displaystyle w_{1}(x)=(1-x^{2})^{-\frac{1}{2}}w\left(x+\mathrm{i}(1-x^{2})^{\frac{1}{2}}\right)}}$ w_{1}(x) = (1-x^{2})^{-\frac{1}{2}}w\left(x+\iunit(1-x^{2})^{\frac{1}{2}}\right)		w[1](x) = (1 - (x)^(2))^(-(1)/(2))* w(x + I*(1 - (x)^(2))^((1)/(2)))	Subscript[w, 1][x] == (1 - (x)^(2))^(-Divide[1,2])* w[x + I*(1 - (x)^(2))^(Divide[1,2])]	Failure	Failure	Failed [300 / 300] Result: 1.128217713+1.045869600*I Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I} Result: -.9208203932+1.594907706*I Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.1282177124267212, 1.0458696000777863] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.9208203932499366, 1.5949077057544443] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
18.33#Ex2	${\displaystyle{\displaystyle w_{2}(x)=(1-x^{2})^{\frac{1}{2}}w\left(x+\mathrm{i}(1-x^{2})^{\frac{1}{2}}\right)}}$ w_{2}(x) = (1-x^{2})^{\frac{1}{2}}w\left(x+\iunit(1-x^{2})^{\frac{1}{2}}\right)		w[2](x) = (1 - (x)^(2))^((1)/(2))* w(x + I*(1 - (x)^(2))^((1)/(2)))	Subscript[w, 2][x] == (1 - (x)^(2))^(Divide[1,2])* w[x + I*(1 - (x)^(2))^(Divide[1,2])]	Failure	Failure	Failed [300 / 300] Result: 1.512563597+.3801630000*I Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[2] = 1/2*3^(1/2)+1/2*I} Result: -.5364745086+.9292011060*I Test Values: {w = 1/2*3^(1/2)+1/2*I, x = 3/2, w[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.5125635972390792, 0.38016299990276686] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.5364745084375786, 0.9292011055794249] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[w, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
18.33.E10	${\displaystyle{\displaystyle z^{-n}\phi_{2n}(z)={A_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+B_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)}}}$ z^{-n}\phi_{2n}(z) = {A_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+B_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)}		(z)^(- n)* phi[2*n](z) = A[n]*p[n]*((1)/(2)*(z + (z)^(- 1)))+ B[n]*(z - (z)^(- 1))*q[n - 1]*((1)/(2)*(z + (z)^(- 1)))	(z)^(- n)* Subscript[\[Phi], 2*n][z] == Subscript[A, n]*Subscript[p, n]*(Divide[1,2]*(z + (z)^(- 1)))+ Subscript[B, n]*(z - (z)^(- 1))*Subscript[q, n - 1]*(Divide[1,2]*(z + (z)^(- 1)))	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33.E11	${\displaystyle{\displaystyle z^{-n+1}\phi_{2n-1}(z)={C_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+D_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)}}}$ z^{-n+1}\phi_{2n-1}(z) = {C_{n}p_{n}\left(\tfrac{1}{2}(z+z^{-1})\right)+D_{n}(z-z^{-1})q_{n-1}\left(\tfrac{1}{2}(z+z^{-1})\right)}		(z)^(- n + 1)* phi[2*n - 1](z) = C[n]*p[n]*((1)/(2)*(z + (z)^(- 1)))+ D[n]*(z - (z)^(- 1))*q[n - 1]*((1)/(2)*(z + (z)^(- 1)))	(z)^(- n + 1)* Subscript[\[Phi], 2*n - 1][z] == Subscript[C, n]*Subscript[p, n]*(Divide[1,2]*(z + (z)^(- 1)))+ Subscript[D, n]*(z - (z)^(- 1))*Subscript[q, n - 1]*(Divide[1,2]*(z + (z)^(- 1)))	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33#Ex3	${\displaystyle{\displaystyle\phi_{n}(z)=z^{n}}}$ \phi_{n}(z) = z^{n}		phi[n](z) = (z)^(n)	Subscript[\[Phi], n][z] == (z)^(n)	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33#Ex4	${\displaystyle{\displaystyle w(z)=1}}$ w(z) = 1		w(z) = 1	w[z] == 1	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33.E13	${\displaystyle{\displaystyle\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+1\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}}}$ \phi_{n}(z) = \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+1}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\,(n-\ell)!}\,z^{\ell}		phi[n](z) = sum((pochhammer(lambda + 1, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*(z)^(ell), ell = 0..n)	Subscript[\[Phi], n][z] == Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Aborted	Failure	Failed [299 / 300] Result: -1.732050808-1.000000000*I Test Values: {lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, n = 1} Result: -.9330127026-4.482050809*I Test Values: {lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, phi[n] = 1/2*3^(1/2)+1/2*I, n = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-1.7320508075688772, -1.0] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.9330127018922204, -4.482050807568885] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
18.33.E13	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{{\left(\lambda+1\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}=\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda+1\atop-\lambda-n+1};z\right)}}$ \sum_{\ell=0}^{n}\frac{\Pochhammersym{\lambda+1}{\ell}\Pochhammersym{\lambda}{n-\ell}}{\ell!\,(n-\ell)!}\,z^{\ell} = \frac{\Pochhammersym{\lambda}{n}}{n!}\genhyperF{2}{1}@@{-n,\lambda+1}{-\lambda-n+1}{z}		sum((pochhammer(lambda + 1, ell)*pochhammer(lambda, n - ell))/(factorial(ell)*factorial(n - ell))*(z)^(ell), ell = 0..n) = (pochhammer(lambda, n))/(factorial(n))*hypergeom([- n , lambda + 1], [- lambda - n + 1], z)	Sum[Divide[Pochhammer[\[Lambda]+ 1, \[ScriptL]]*Pochhammer[\[Lambda], n - \[ScriptL]],(\[ScriptL])!*(n - \[ScriptL])!]*(z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[Pochhammer[\[Lambda], n],(n)!]*HypergeometricPFQ[{- n , \[Lambda]+ 1}, {- \[Lambda]- n + 1}, z]	Aborted	Successful	Successful [Tested: 0]	Failed [21 / 210] Result: Indeterminate Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, -2]} ... skip entries to safe data
18.33#Ex5	${\displaystyle{\displaystyle w(z)=\left(1-\tfrac{1}{2}(z+z^{-1})\right)^{\lambda}}}$ w(z) = \left(1-\tfrac{1}{2}(z+z^{-1})\right)^{\lambda}		w(z) = (1 -(1)/(2)*(z + (z)^(- 1)))^(lambda)	w[z] == (1 -Divide[1,2]*(z + (z)^(- 1)))^\[Lambda]	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33#Ex6	${\displaystyle{\displaystyle w_{1}(x)=(1-x)^{\lambda-\frac{1}{2}}(1+x)^{-\frac{1}{2}}}}$ w_{1}(x) = (1-x)^{\lambda-\frac{1}{2}}(1+x)^{-\frac{1}{2}}		w[1](x) = (1 - x)^(lambda -(1)/(2))*(1 + x)^(-(1)/(2))	Subscript[w, 1][x] == (1 - x)^(\[Lambda]-Divide[1,2])*(1 + x)^(-Divide[1,2])	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33#Ex7	${\displaystyle{\displaystyle w_{2}(x)=(1-x)^{\lambda+\frac{1}{2}}(1+x)^{\frac{1}{2}}}}$ w_{2}(x) = (1-x)^{\lambda+\frac{1}{2}}(1+x)^{\frac{1}{2}}	${\displaystyle{\displaystyle\lambda>-\tfrac{1}{2}}}$	w[2](x) = (1 - x)^(lambda +(1)/(2))*(1 + x)^((1)/(2))	Subscript[w, 2][x] == (1 - x)^(\[Lambda]+Divide[1,2])*(1 + x)^(Divide[1,2])	Skipped - no semantic math	Skipped - no semantic math	-	-
18.33.E15	${\displaystyle{\displaystyle\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{\left(aq^{2};q^{2}\right)_{\ell}\left(a;q^{2}\right)_{n-\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q^{2};q^{2}\right)_{n-\ell}}(q^{-1}z)^{\ell}}}$ \phi_{n}(z) = \sum_{\ell=0}^{n}\frac{\qPochhammer{aq^{2}}{q^{2}}{\ell}\qPochhammer{a}{q^{2}}{n-\ell}}{\qPochhammer{q^{2}}{q^{2}}{\ell}\qPochhammer{q^{2}}{q^{2}}{n-\ell}}(q^{-1}z)^{\ell}		phi[n](z) = sum((QPochhammer(a*(q)^(2), (q)^(2), ell)*QPochhammer(a, (q)^(2), n - ell))/(QPochhammer((q)^(2), (q)^(2), ell)*QPochhammer((q)^(2), (q)^(2), n - ell))*((q)^(- 1)* z)^(ell), ell = 0..n)	Subscript[\[Phi], n][z] == Sum[Divide[QPochhammer[a*(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[a, (q)^(2), n - \[ScriptL]],QPochhammer[(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[(q)^(2), (q)^(2), n - \[ScriptL]]]*((q)^(- 1)* z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None]	Aborted	Aborted	Error	Skipped - Because timed out
18.33.E15	${\displaystyle{\displaystyle\sum_{\ell=0}^{n}\frac{\left(aq^{2};q^{2}\right)_{\ell}\left(a;q^{2}\right)_{n-\ell}}{\left(q^{2};q^{2}\right)_{\ell}\left(q^{2};q^{2}\right)_{n-\ell}}(q^{-1}z)^{\ell}=\frac{\left(a;q^{2}\right)_{n}}{\left(q^{2};q^{2}\right)_{n}}{{}_{2}\phi_{1}}\left({aq^{2},q^{-2n}\atop a^{-1}q^{2-2n}};q^{2},\frac{qz}{a}\right)}}$ \sum_{\ell=0}^{n}\frac{\qPochhammer{aq^{2}}{q^{2}}{\ell}\qPochhammer{a}{q^{2}}{n-\ell}}{\qPochhammer{q^{2}}{q^{2}}{\ell}\qPochhammer{q^{2}}{q^{2}}{n-\ell}}(q^{-1}z)^{\ell} = \frac{\qPochhammer{a}{q^{2}}{n}}{\qPochhammer{q^{2}}{q^{2}}{n}}\qgenhyperphi{2}{1}@@{aq^{2},q^{-2n}}{a^{-1}q^{2-2n}}{q^{2}}{\frac{qz}{a}}		Error	Sum[Divide[QPochhammer[a*(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[a, (q)^(2), n - \[ScriptL]],QPochhammer[(q)^(2), (q)^(2), \[ScriptL]]*QPochhammer[(q)^(2), (q)^(2), n - \[ScriptL]]]*((q)^(- 1)* z)^\[ScriptL], {\[ScriptL], 0, n}, GenerateConditions->None] == Divide[QPochhammer[a, (q)^(2), n],QPochhammer[(q)^(2), (q)^(2), n]]*QHypergeometricPFQ[{a*(q)^(2), (q)^(- 2*n)},{(a)^(- 1)* (q)^(2 - 2*n)},(q)^(2),Divide[q*z,a]]	Missing Macro Error	Aborted	Skip - symbolical successful subtest	Skipped - Because timed out