Combinatorial Analysis - 26.10 Integer Partitions: Other Restrictions

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
26.10.E2	${\displaystyle{\displaystyle\prod_{j=1}^{\infty}(1+q^{j})=\prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}}}$ \prod_{j=1}^{\infty}(1+q^{j}) = \prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}		product(1 + (q)^(j), j = 1..infinity) = product((1)/(1 - (q)^(2*j - 1)), j = 1..infinity)	Product[1 + (q)^(j), {j, 1, Infinity}, GenerateConditions->None] == Product[Divide[1,1 - (q)^(2*j - 1)], {j, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [1 / 10] Result: DirectedInfinity[] Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
26.10.E3	${\displaystyle{\displaystyle\sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}q^{m(m+1)/2}x^{m}=\prod_{j=1}^{k}(1+x\,q^{j})}}$ \sum_{m=0}^{k}\qbinom{k}{m}{q}q^{m(m+1)/2}x^{m} = \prod_{j=1}^{k}(1+x\,q^{j})	${\displaystyle{\displaystyle|x|<1}}$	sum(QBinomial(k, m, q)*(q)^(m*(m + 1)/2)* (x)^(m), m = 0..k) = product(1 + x*(q)^(j), j = 1..k)	Sum[QBinomial[k,m,q]*(q)^(m*(m + 1)/2)* (x)^(m), {m, 0, k}, GenerateConditions->None] == Product[1 + x*(q)^(j), {j, 1, k}, GenerateConditions->None]	Failure	Failure	Error	Successful [Tested: 30]