Orthogonal Polynomials - 18.19 Hahn Class: Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.19.E1 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 - -
18.19.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(z;a,b,\conj{a},\conj{b}) = \EulerGamma@{a+iz}\EulerGamma@{b+iz}\EulerGamma@{\conj{a}-iz}\EulerGamma@{\conj{b}-iz}}
w(z;a,b,\conj{a},\conj{b}) = \EulerGamma@{a+iz}\EulerGamma@{b+iz}\EulerGamma@{\conj{a}-iz}\EulerGamma@{\conj{b}-iz}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
w(z ; a , b , conjugate(a), conjugate(b)) = GAMMA(a + I*z)*GAMMA(b + I*z)*GAMMA(conjugate(a)- I*z)*GAMMA(conjugate(b)- I*z)

w[z ; a , b , Conjugate[a], Conjugate[b]] == Gamma[a + I*z]*Gamma[b + I*z]*Gamma[Conjugate[a]- I*z]*Gamma[Conjugate[b]- I*z]
Translation Error Translation Error - -
18.19.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(x) = w(x;a,b,\conj{a},\conj{b})}
w(x) = w(x;a,b,\conj{a},\conj{b})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
w(x) = w(x ; a , b , conjugate(a), conjugate(b))

w[x] == w[x ; a , b , Conjugate[a], Conjugate[b]]
Translation Error Translation Error - -
18.19.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(x;a,b,\conj{a},\conj{b}) = |\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}|^{2}}
w(x;a,b,\conj{a},\conj{b}) = |\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}|^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(a+\iunit x)} > 0, \realpart@@{(b+\iunit x)} > 0} 
w(x ; a , b , conjugate(a), conjugate(b)) = (abs(GAMMA(a + I*x)*GAMMA(b + I*x)))^(2)

w[x ; a , b , Conjugate[a], Conjugate[b]] == (Abs[Gamma[a + I*x]*Gamma[b + I*x]])^(2)
Translation Error Translation Error - -
18.19.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k_{n} = \frac{\Pochhammersym{n+2\realpart@{a+b}-1}{n}}{n!}}
k_{n} = \frac{\Pochhammersym{n+2\realpart@{a+b}-1}{n}}{n!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
k[n] = (pochhammer(n + 2*Re(a + b)- 1, n))/(factorial(n))

Subscript[k, n] == Divide[Pochhammer[n + 2*Re[a + b]- 1, n],(n)!]
Failure Failure
Failed [298 / 300]
Result: 6.866025404+.5000000000*I
Test Values: {a = -3/2, b = -3/2, k[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -9.133974596+.5000000000*I
Test Values: {a = -3/2, b = -3/2, k[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [298 / 300]
Result: Complex[6.866025403784438, 0.49999999999999994]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[Subscript[k, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-9.13397459621556, 0.49999999999999994]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 2], Rule[Subscript[k, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
18.19.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w^{(\lambda)}(z;\phi) = \EulerGamma@{\lambda+iz}\EulerGamma@{\lambda-iz}e^{(2\phi-\pi)z}}
w^{(\lambda)}(z;\phi) = \EulerGamma@{\lambda+iz}\EulerGamma@{\lambda-iz}e^{(2\phi-\pi)z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\lambda+\iunit z)} > 0, \realpart@@{(\lambda-\iunit z)} > 0} 
(w(z ; phi))^(lambda) = GAMMA(lambda + I*z)*GAMMA(lambda - I*z)*exp((2*phi - Pi)*z)

(w[z ; \[Phi]])^(\[Lambda]) == Gamma[\[Lambda]+ I*z]*Gamma[\[Lambda]- I*z]*Exp[(2*\[Phi]- Pi)*z]
Translation Error Translation Error - -
18.19.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(x) = w^{(\lambda)}(x;\phi)}
w(x) = w^{(\lambda)}(x;\phi)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda > 0, 0 < \phi, \phi < \pi} 
w(x) = (w(x ; phi))^(lambda)

w[x] == (w[x ; \[Phi]])^(\[Lambda])
Translation Error Translation Error - -
18.19.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w^{(\lambda)}(x;\phi) = \left|\EulerGamma@{\lambda+\iunit x}\right|^{2}e^{(2\phi-\pi)x}}
w^{(\lambda)}(x;\phi) = \left|\EulerGamma@{\lambda+\iunit x}\right|^{2}e^{(2\phi-\pi)x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lambda > 0, 0 < \phi, \phi < \pi, \realpart@@{(\lambda+\iunit x)} > 0} 
(w(x ; phi))^(lambda) = (abs(GAMMA(lambda + I*x)))^(2)* exp((2*phi - Pi)*x)

(w[x ; \[Phi]])^(\[Lambda]) == (Abs[Gamma[\[Lambda]+ I*x]])^(2)* Exp[(2*\[Phi]- Pi)*x]
Translation Error Translation Error - -
18.19#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle k_{n} = \frac{(2\sin@@{\phi})^{n}}{n!}}
k_{n} = \frac{(2\sin@@{\phi})^{n}}{n!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
k[n] = ((2*sin(phi))^(n))/(factorial(n))

Subscript[k, n] == Divide[(2*Sin[\[Phi]])^(n),(n)!]
Failure Failure
Failed [300 / 300]
Result: -.8519352650-.1751929262*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: -.3817262820-.6599548910*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.851935264815837, -0.17519292644574008]
Test Values: {Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.3817262816831334, -0.6599548913509004]
Test Values: {Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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