On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: An implementation of Radu's algorithm

Abstract

In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg, in which the generating functions of a partition function over a given set of arithmetic progressions are expressed in terms of Dedekind eta quotients over a given congruence subgroup. These identities include the famous results by Ramanujan which provide a witness to the divisibility properties of p(5n+4), p(7n+5). We give an implementation of this algorithm using Mathematica. The basic theory is first described, and an outline of the algorithm is briefly given, in order to describe the functionality and utility of our package. We thereafter give multiple examples of applications to recent work in partition theory. In many cases we have used our package to derive alternate proofs of various identities or congruences; in other cases we have improved previously established identities.