q -Hypergeometric and Related Functions - 18.1 Notation

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	-	-