Finding Modular Functions for Ramanujan-Type Identities

Abstract

This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for a(mn + t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n + 6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions \( \mathit(\overline{p}) \)(5n + 2) and \( \mathit(\overline{p}) \)(5n + 3) and Andrews–Paule’s broken 2-diamond partition functions \( \triangle_{2} \)(25n+14) and \( \triangle_{2} \)(25n+24). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews’ singular overpartition functions \( \mathit(\overline{Q}_{3,1}) \)(9n + 3) and \( \mathit(\overline{Q}_{3,1}) \)(9n + 6) due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.