26.9: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/26.9.E4 26.9.E4] || [[Item:Q7877|<math>\qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}</syntaxhighlight> || <math>n \geq 0</math> || <syntaxhighlight lang=mathematica>QBinomial(m, n, q) = product((1 - (q)^(m - n + j))/(1 - (q)^(j)), j = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QBinomial[m,n,q] == Product[Divide[1 - (q)^(m - n + j),1 - (q)^(j)], {j, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [32 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
| [https://dlmf.nist.gov/26.9.E4 26.9.E4] || <math qid="Q7877">\qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}</syntaxhighlight> || <math>n \geq 0</math> || <syntaxhighlight lang=mathematica>QBinomial(m, n, q) = product((1 - (q)^(m - n + j))/(1 - (q)^(j)), j = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QBinomial[m,n,q] == Product[Divide[1 - (q)^(m - n + j),1 - (q)^(j)], {j, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [32 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[m, 2], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[m, 2], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/26.9.E5 26.9.E5] || [[Item:Q7878|<math>\prod_{j=1}^{k}\frac{1}{1-q^{j}} = 1+\sum_{m=1}^{\infty}\qbinom{k+m-1}{m}{q}q^{m}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\prod_{j=1}^{k}\frac{1}{1-q^{j}} = 1+\sum_{m=1}^{\infty}\qbinom{k+m-1}{m}{q}q^{m}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>product((1)/(1 - (q)^(j)), j = 1..k) = 1 + sum(QBinomial(k + m - 1, m, q)*(q)^(m), m = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Product[Divide[1,1 - (q)^(j)], {j, 1, k}, GenerateConditions->None] == 1 + Sum[QBinomial[k + m - 1,m,q]*(q)^(m), {m, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
| [https://dlmf.nist.gov/26.9.E5 26.9.E5] || <math qid="Q7878">\prod_{j=1}^{k}\frac{1}{1-q^{j}} = 1+\sum_{m=1}^{\infty}\qbinom{k+m-1}{m}{q}q^{m}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\prod_{j=1}^{k}\frac{1}{1-q^{j}} = 1+\sum_{m=1}^{\infty}\qbinom{k+m-1}{m}{q}q^{m}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>product((1)/(1 - (q)^(j)), j = 1..k) = 1 + sum(QBinomial(k + m - 1, m, q)*(q)^(m), m = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Product[Divide[1,1 - (q)^(j)], {j, 1, k}, GenerateConditions->None] == 1 + Sum[QBinomial[k + m - 1,m,q]*(q)^(m), {m, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
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| [https://dlmf.nist.gov/26.9.E7 26.9.E7] || [[Item:Q7880|<math>1+\sum_{k=1}^{\infty}\qbinom{m+k}{k}{q}x^{k} = \prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1+\sum_{k=1}^{\infty}\qbinom{m+k}{k}{q}x^{k} = \prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>1 + sum(QBinomial(m + k, k, q)*(x)^(k), k = 1..infinity) = product((1)/(1 - x*(q)^(j)), j = 0..m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 + Sum[QBinomial[m + k,k,q]*(x)^(k), {k, 1, Infinity}, GenerateConditions->None] == Product[Divide[1,1 - x*(q)^(j)], {j, 0, m}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
| [https://dlmf.nist.gov/26.9.E7 26.9.E7] || <math qid="Q7880">1+\sum_{k=1}^{\infty}\qbinom{m+k}{k}{q}x^{k} = \prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1+\sum_{k=1}^{\infty}\qbinom{m+k}{k}{q}x^{k} = \prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>1 + sum(QBinomial(m + k, k, q)*(x)^(k), k = 1..infinity) = product((1)/(1 - x*(q)^(j)), j = 0..m)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 + Sum[QBinomial[m + k,k,q]*(x)^(k), {k, 1, Infinity}, GenerateConditions->None] == Product[Divide[1,1 - x*(q)^(j)], {j, 0, m}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
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Latest revision as of 12:06, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
26.9.E4 [ m n ] q = ∏ j = 1 n 1 - q m - n + j 1 - q j q-binomial 𝑚 𝑛 𝑞 superscript subscript product 𝑗 1 𝑛 1 superscript 𝑞 𝑚 𝑛 𝑗 1 superscript 𝑞 𝑗 {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}=\prod_{j=1}^{n}% \frac{1-q^{m-n+j}}{1-q^{j}}}}
\qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}		 n ≥ 0 𝑛 0 {\displaystyle{\displaystyle n\geq 0}} 
QBinomial(m, n, q) = product((1 - (q)^(m - n + j))/(1 - (q)^(j)), j = 1..n)

QBinomial[m,n,q] == Product[Divide[1 - (q)^(m - n + j),1 - (q)^(j)], {j, 1, n}, GenerateConditions->None]
Failure Failure Error
Failed [32 / 90]
Result: DirectedInfinity[]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: DirectedInfinity[]
Test Values: {Rule[m, 2], Rule[n, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
26.9.E5 ∏ j = 1 k 1 1 - q j = 1 + ∑ m = 1 ∞ [ k + m - 1 m ] q ⁢ q m superscript subscript product 𝑗 1 𝑘 1 1 superscript 𝑞 𝑗 1 superscript subscript 𝑚 1 q-binomial 𝑘 𝑚 1 𝑚 𝑞 superscript 𝑞 𝑚 {\displaystyle{\displaystyle\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1+\sum_{m=1}^{% \infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m}}}
\prod_{j=1}^{k}\frac{1}{1-q^{j}} = 1+\sum_{m=1}^{\infty}\qbinom{k+m-1}{m}{q}q^{m}		 

product((1)/(1 - (q)^(j)), j = 1..k) = 1 + sum(QBinomial(k + m - 1, m, q)*(q)^(m), m = 1..infinity)

Product[Divide[1,1 - (q)^(j)], {j, 1, k}, GenerateConditions->None] == 1 + Sum[QBinomial[k + m - 1,m,q]*(q)^(m), {m, 1, Infinity}, GenerateConditions->None]
Failure Aborted Error Skipped - Because timed out
26.9.E7 1 + ∑ k = 1 ∞ [ m + k k ] q ⁢ x k = ∏ j = 0 m 1 1 - x ⁢ q j 1 superscript subscript 𝑘 1 q-binomial 𝑚 𝑘 𝑘 𝑞 superscript 𝑥 𝑘 superscript subscript product 𝑗 0 𝑚 1 1 𝑥 superscript 𝑞 𝑗 {\displaystyle{\displaystyle 1+\sum_{k=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{m+k}% {k}_{q}x^{k}=\prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}}}
1+\sum_{k=1}^{\infty}\qbinom{m+k}{k}{q}x^{k} = \prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}		 

1 + sum(QBinomial(m + k, k, q)*(x)^(k), k = 1..infinity) = product((1)/(1 - x*(q)^(j)), j = 0..m)

1 + Sum[QBinomial[m + k,k,q]*(x)^(k), {k, 1, Infinity}, GenerateConditions->None] == Product[Divide[1,1 - x*(q)^(j)], {j, 0, m}, GenerateConditions->None]
Failure Aborted Error Skipped - Because timed out
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