Combinatorial Analysis - 26.9 Integer Partitions:

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
26.9.E4	${\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{m}{n}_{q}=\prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}}}$ \qbinom{m}{n}{q} = \prod_{j=1}^{n}\frac{1-q^{m-n+j}}{1-q^{j}}	${\displaystyle{\displaystyle n\geq 0}}$	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	${\displaystyle{\displaystyle\prod_{j=1}^{k}\frac{1}{1-q^{j}}=1+\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}{m}_{q}q^{m}}}$ \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	${\displaystyle{\displaystyle 1+\sum_{k=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{q}x^{k}=\prod_{j=0}^{m}\frac{1}{1-x\,q^{j}}}}$ 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