Elementary proofs in the theory of partitions (Q1131280)

Let \(k>1\) be an integer, and let \(a_1, a_2, \ldots, a_s\) be \(s\) distinct positive integers not exceeding \(k\). The set of nonnegative integers \(\nu\) such that \(\nu\equiv a_i \pmod k\) for some \(i\), \(1\leq i\leq s\) is called a congruence set modulo \(k\), and is denoted by \(a\). Let \(F(x; a)= \prod_{\nu\in a} (1-x^\nu)^{-1} = \sum p_n(a) x^n\), so that \(p_n(a)\) is the number of partitions of \(n\) into summands from \(a\). Let \(b\) be another congruence set modulo \(k\), based on \(b_1, b_2, \ldots, b_t\). The author proves a number of theorems about such congruence sets, among which the following is typical: Suppose that \(s = t\) and that \(\sum_{i=1}^t a_i = \sum_{i=1}^t b_i\). Then \[ \lim_{x\to 1-} \frac{F(x;a)}{F(x;b)} = \lim_{x\to\infty} \frac{p_n(a)}{p_n(b)}=\prod_{j=1}^t\left\{\frac{\Gamma(a_j/k)}{\Gamma(b_j/k)}\right\}. \]