A remarkable sequence derived from Euler products (Q2738241)

The author introduces a sequence of products NEWLINE\[NEWLINEa_n(\alpha)= \prod _{\substack{ m=0\\ m\neq n}}^\infty \frac{1} {1-\alpha^{m-n}}NEWLINE\]NEWLINE where \(0< \alpha< 1\). The function \(a_0(\alpha)\) is Euler's generating product for the partition function \(p(n)\). The author derives many interesting properties of these products, two of the simplest being the equation NEWLINE\[NEWLINEa_0(\alpha)= \sum_{n=1}^\infty a_{n-1} (\alpha)/ (1-\alpha^n)NEWLINE\]NEWLINE and the recursion \(a_n(\alpha)= a_{n-1}(\alpha) \alpha^n/ (1-\alpha^n)\), which leads to an efficient method for calculating \(a_n(\alpha)\) in terms of \(a_0(\alpha)\). He also shows that the \(a_n(\alpha)\) occur in several diverse areas of mathematics.