Several $q$-series identities from the Euler expansions of $(a;q)_{\infty }$ and $\frac{1}{(a;q)_{\infty }}$

Zhizheng Zhang; Yang, Jizhen

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Introduction. The problem of determining the formula for $P_{S} (n)$ , the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, $h_{s ₁}, . . ., h_{s_{k}}$ , of the equation hs₁ s₁ + ... + hsk sk = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient of $x ⁿ i n$ [(1-xs₁)... (1-xsk)]-1, expressions for which they derived. Wright [5] indicated a simpler method by which to find part...