Orthogonal Polynomials - 18.22 Hahn Class: Recurrence Relations and

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.22.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -xp_{n}(x) = A_{n}p_{n+1}(x)-\left(A_{n}+C_{n}\right)p_{n}(x)+C_{n}p_{n-1}(x)}
-xp_{n}(x) = A_{n}p_{n+1}(x)-\left(A_{n}+C_{n}\right)p_{n}(x)+C_{n}p_{n-1}(x)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
- xp[n](x) = A[n]*p[n + 1](x)-(A[n]+((n*(n + alpha + beta + N + 1)*(n + beta))/((2*n + alpha + beta)*(2*n + alpha + beta + 1))))*p[n](x)+((n*(n + alpha + beta + N + 1)*(n + beta))/((2*n + alpha + beta)*(2*n + alpha + beta + 1)))*p[n - 1](x)
 
- Subscript[xp, n][x] == Subscript[A, n]*Subscript[p, n + 1][x]-(Subscript[A, n]+(Divide[n*(n + \[Alpha]+ \[Beta]+ N + 1)*(n + \[Beta]),(2*n + \[Alpha]+ \[Beta])*(2*n + \[Alpha]+ \[Beta]+ 1)]))*Subscript[p, n][x]+(Divide[n*(n + \[Alpha]+ \[Beta]+ N + 1)*(n + \[Beta]),(2*n + \[Alpha]+ \[Beta])*(2*n + \[Alpha]+ \[Beta]+ 1)])*Subscript[p, n - 1][x]
 Skipped - no semantic math Skipped - no semantic math - -
18.22.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q_{n}(x) = \ifrac{\contHahnpolyp{n}@{x}{a}{b}{\conj{a}}{\conj{b}}}{\contHahnpolyp{n}@{\iunit a}{a}{b}{\conj{a}}{\conj{b}}}}
q_{n}(x) = \ifrac{\contHahnpolyp{n}@{x}{a}{b}{\conj{a}}{\conj{b}}}{\contHahnpolyp{n}@{\iunit a}{a}{b}{\conj{a}}{\conj{b}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Subscript[q, n][x] == Divide[I^(n)*Divide[Pochhammer[a + Conjugate[a], n]*Pochhammer[a + Conjugate[b], n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[a + b] - 1, a + I*(x)}, {a + Conjugate[a], a + Conjugate[b]}, 1],I^(n)*Divide[Pochhammer[a + Conjugate[a], n]*Pochhammer[a + Conjugate[b], n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[a + b] - 1, a + I*(I*a)}, {a + Conjugate[a], a + Conjugate[b]}, 1]]
Missing Macro Error Missing Macro Error - -
18.22.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (n+1)p_{n+1}(x) = 2\left(x\sin@@{\phi}+(n+\lambda)\cos@@{\phi}\right)p_{n}(x)-(n+2\lambda-1)p_{n-1}(x)}
(n+1)p_{n+1}(x) = 2\left(x\sin@@{\phi}+(n+\lambda)\cos@@{\phi}\right)p_{n}(x)-(n+2\lambda-1)p_{n-1}(x)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(n + 1)*p[n + 1](x) = 2*(x*sin(phi)+(n + lambda)*cos(phi))*p[n](x)-(n + 2*lambda - 1)*p[n - 1](x)

(n + 1)*Subscript[p, n + 1][x] == 2*(x*Sin[\[Phi]]+(n + \[Lambda])*Cos[\[Phi]])*Subscript[p, n][x]-(n + 2*\[Lambda]- 1)*Subscript[p, n - 1][x]
Failure Failure
Failed [300 / 300]
Result: -3.110426782-.517373007*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, p[n] = 1/2*3^(1/2)+1/2*I, p[n-1] = 1/2*3^(1/2)+1/2*I, p[n+1] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -3.005781337+.918117648*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, p[n] = 1/2*3^(1/2)+1/2*I, p[n-1] = 1/2*3^(1/2)+1/2*I, p[n+1] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-3.110426781913132, -0.5173730098941742]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.005781335086172, 0.9181176450774369]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, Plus[-1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.22.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x)p_{n}(x+1)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-1)-n(n+\alpha+\beta+1)p_{n}(x) = 0}
A(x)p_{n}(x+1)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-1)-n(n+\alpha+\beta+1)p_{n}(x) = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A(x)* p[n](x + 1)-(A(x)+(x*(x - beta - N - 1)))*p[n](x)+(x*(x - beta - N - 1))*p[n](x - 1)- n*(n + alpha + beta + 1)*p[n](x) = 0
 
A[x]* Subscript[p, n][x + 1]-(A[x]+(x*(x - \[Beta]- N - 1)))*Subscript[p, n][x]+(x*(x - \[Beta]- N - 1))*Subscript[p, n][x - 1]- n*(n + \[Alpha]+ \[Beta]+ 1)*Subscript[p, n][x] == 0
 Skipped - no semantic math Skipped - no semantic math - -
18.22.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x)p_{n}(x+1)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-1)+\lambda_{n}p_{n}(x) = 0}
A(x)p_{n}(x+1)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-1)+\lambda_{n}p_{n}(x) = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A(x)* p[n](x + 1)-(A(x)+ C(x))*p[n](x)+ C(x)* p[n](x - 1)+ lambda[n]*p[n](x) = 0
 
A[x]* Subscript[p, n][x + 1]-(A[x]+ C[x])*Subscript[p, n][x]+ C[x]* Subscript[p, n][x - 1]+ Subscript[\[Lambda], n]*Subscript[p, n][x] == 0
 Skipped - no semantic math Skipped - no semantic math - -
18.22.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle p_{n}(x) = \contHahnpolyp{n}@{x}{a}{b}{\conj{a}}{\conj{b}}}
p_{n}(x) = \contHahnpolyp{n}@{x}{a}{b}{\conj{a}}{\conj{b}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Subscript[p, n][x] == I^(n)*Divide[Pochhammer[a + Conjugate[a], n]*Pochhammer[a + Conjugate[b], n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[a + b] - 1, a + I*(x)}, {a + Conjugate[a], a + Conjugate[b]}, 1]
Missing Macro Error Missing Macro Error - -
18.22.E14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+n(n+2\realpart@{a+b}-1)p_{n}(x) = 0}
A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+n(n+2\realpart@{a+b}-1)p_{n}(x) = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A(x)* p[n](x + I)-(A(x)+((x - I*a)*(x - I*b)))*p[n](x)+((x - I*a)*(x - I*b))*p[n](x - I)+ n*(n + 2*Re(a + b)- 1)*p[n](x) = 0

A[x]* Subscript[p, n][x + I]-(A[x]+((x - I*a)*(x - I*b)))*Subscript[p, n][x]+((x - I*a)*(x - I*b))*Subscript[p, n][x - I]+ n*(n + 2*Re[a + b]- 1)*Subscript[p, n][x] == 0
Failure Failure
Failed [300 / 300]
Result: -5.196152425-1.499999999*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, x = 3/2, p[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -10.39230485-4.499999999*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, x = 3/2, p[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-5.196152422706632, -1.5]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-10.392304845413264, -4.5]
Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.22.E17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+2n\sin@@{\phi}\,p_{n}(x) = 0}
A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+2n\sin@@{\phi}\,p_{n}(x) = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A(x)* p[n](x + I)-(A(x)+(exp(- I*phi)*(x - I*lambda)))*p[n](x)+(exp(- I*phi)*(x - I*lambda))*p[n](x - I)+ 2*n*sin(phi)*p[n](x) = 0

A[x]* Subscript[p, n][x + I]-(A[x]+(Exp[- I*\[Phi]]*(x - I*\[Lambda])))*Subscript[p, n][x]+(Exp[- I*\[Phi]]*(x - I*\[Lambda]))*Subscript[p, n][x - I]+ 2*n*Sin[\[Phi]]*Subscript[p, n][x] == 0
Failure Failure
Failed [300 / 300]
Result: -2.025869520+.288999556*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, p[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.300567841+2.454571398*I
Test Values: {A = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, p[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-2.025869520811228, 0.28899955435496594]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.3005678430800254, 2.4545713959415254]
Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[x, 1.5], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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