DLMF:26.9.E7 (Q7880)

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DLMF:26.9.E7
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    ∑ m , n = 0 ∞ p k ⁡ ( ≤ m , n ) ⁢ x k ⁢ q n = 1 + ∑ k = 1 ∞ [ m + k k ] q ⁢ x k = ∏ j = 0 m 1 1 - x ⁢ q j . superscript subscript 𝑚 𝑛 0 restricted-partitions-P-at-most 𝑘 absent 𝑚 𝑛 superscript 𝑥 𝑘 superscript 𝑞 𝑛 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\sum_{m,n=0}^{\infty}p_{k}\left(\leq m,n\right)x^{% k}q^{n}=1+\sum_{k=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{q}x^{k}=\prod_{j% =0}^{m}\frac{1}{1-x\,q^{j}}.}}
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    DLMF:26.9.E7
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