Finding Modular Functions for Ramanujan-Type Identities

TL;DR

This paper develops an algorithm to find Ramanujan-type identities for a class of partition functions defined by eta-quotients, successfully deriving identities for various overpartition and broken partition functions.

Contribution

It introduces a new algorithm based on transformation laws and Radu's methods to discover Ramanujan-type identities for complex partition functions.

Findings

Derived identities for overpartition functions n+2 and n+3.

Established identities for broken 2-diamond partition functions.

Extended the method to general classes of partition functions.

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 $\overline{p}(5n+2)$ and $\overline{p}(5n+3)$ 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 $\overline{Q}_{3,1}(9n+3)$ and $ \overline{Q}_{3,1}(9n+6)$ due to Shen, the $2$-dissection formulas of Ramanujan and the $8$-dissection formulas due to Hirschhorn.