Product signatures

A power series \[ 1 + \sum_{n=1}^\infty r(n) q^n ,\] where the \( r(n) \) are integers, can be factored into an infinite product of the form \[ \prod_{n=1}^\infty \frac{1}{(1-q)^{a_n}}, \] for integers \( a_1, a_2, a_3, \dots ,\) see Andrews (1976, p. 98, Ex. 2). For the infinite products that arise in Rogers–Ramanujan type identities, the \( a_n \) are periodic functions of \(n\) with respect to a fixed modulus \( m \), i.e. the products are of the form \[ \prod_{i=1}^m \frac{1}{(q^i;q^m)^{a_i}}. \tag{RRP} \] We define the product signature of (RRP) to be \[ m:(a_1, a_2, a_3, \dots, a_m). \] Of course, we choose \( m \) to be the minimal modulus for which such a representation (RRP) exists.

Typesetting tool

This tool will typeset a product of \(q\)-Pochhammer symbols in \(\LaTeX\) from a given product signature.

Modulus: Powers: