Mock theta functions and related combinatorics

TL;DR

This paper explores the combinatorial properties of Ramanujan's mock theta functions, establishing new identities related to partition theory through analytic and combinatorial methods.

Contribution

It introduces Beck-type identities for third order mock theta functions, linking mock modular forms with partition combinatorics in a novel way.

Findings

Established Beck-type identities for mock theta functions

Connected mock theta functions with partition identities

Used combinatorial bijections and analytic proofs

Abstract

In this paper we add to the literature on the combinatorial nature of the mock theta functions, a collection of curious $q$-hypergeometric series introduced by Ramanujan in his last letter to Hardy in 1920, which we now know to be important examples of mock modular forms. Our work is inspired by Beck's conjecture, now a theorem of Andrews, related to Euler's identity: the excess of the number of parts in all partitions of $n$ into odd parts over the number of partitions of $n$ into distinct parts is equal to the number of partitions with only one (possibly repeated) even part and all other parts odd. We establish Beck-type identities associated to partition identities due to Andrews, Dixit, and Yee for the third order mock theta functions $\omega(q), \nu(q)$, and $\phi(q)$. Our proofs are both analytic and combinatorial in nature, and involve mock theta generating functions and combinatorial bijections.