View source for 18.1

{{DISPLAYTITLE:q -Hypergeometric and Related Functions - 18.1 Notation}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/18.1#Ex7 18.1#Ex7] || <math qid="Q5481">\qPochhammer{z}{q}{0} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qPochhammer{z}{q}{0} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QPochhammer(z, q, 0) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>QPochhammer[z, q, 0] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
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| [https://dlmf.nist.gov/18.1#Ex10 18.1#Ex10] || <math qid="Q5484">\qPochhammer{z}{q}{\infty} = \prod_{j=0}^{\infty}(1-zq^{j})</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qPochhammer{z}{q}{\infty} = \prod_{j=0}^{\infty}(1-zq^{j})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QPochhammer(z, q, infinity) = product(1 - z*(q)^(j), j = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QPochhammer[z, q, Infinity] == Product[1 - z*(q)^(j), {j, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [56 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/18.1.E1 18.1.E1] || <math qid="Q5486">\ultrasphpoly{0}{n}@{x} = \frac{2}{n}\ChebyshevpolyT{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ultrasphpoly{0}{n}@{x} = \frac{2}{n}\ChebyshevpolyT{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GegenbauerC(n, 0, x) = (2)/(n)*ChebyshevT(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GegenbauerC[n, 0, x] == Divide[2,n]*ChebyshevT[n, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -6.0
Test Values: {Rule[n, 3], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 0.6666666666666666
Test Values: {Rule[n, 3], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/18.1.E1 18.1.E1] || <math qid="Q5486">\frac{2}{n}\ChebyshevpolyT{n}@{x} = \frac{2(n-1)!}{\Pochhammersym{\tfrac{1}{2}}{n}}\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2}{n}\ChebyshevpolyT{n}@{x} = \frac{2(n-1)!}{\Pochhammersym{\tfrac{1}{2}}{n}}\JacobipolyP{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)/(n)*ChebyshevT(n, x) = (2*factorial(n - 1))/(pochhammer((1)/(2), n))*JacobiP(n, -(1)/(2), -(1)/(2), x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2,n]*ChebyshevT[n, x] == Divide[2*(n - 1)!,Pochhammer[Divide[1,2], n]]*JacobiP[n, -Divide[1,2], -Divide[1,2], x]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
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| [https://dlmf.nist.gov/18.1.E2 18.1.E2] || <math qid="Q5487">\shiftJacobipolyG{n}@{p}{q}{x} = \frac{n!}{\Pochhammersym{n+p}{n}}\JacobipolyP{p-q}{q-1}{n}@{2x-1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\shiftJacobipolyG{n}@{p}{q}{x} = \frac{n!}{\Pochhammersym{n+p}{n}}\JacobipolyP{p-q}{q-1}{n}@{2x-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
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