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arXiv · 1603.04352

Overpartitions related to the mock theta function $\omega(q)$

George E. Andrews·Atul Dixit·Daniel Schultz·Ae Ja Yee
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Abstract

It was recently shown that $q\omega(q)$, where $\omega(q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the smallest part. In this paper, we study the overpartition analogue of $p_{\omega}(n)$, and express its generating function in terms of a ${}_3\phi_{2}$ basic hypergeometric series and an infinite series involving little $q$-Jacobi polynomials. This is accomplished by obtaining a new seven parameter $q$-series identity which generalizes a deep identity due to the first author as well as its generalization by R.P.~Agarwal. We also derive two interesting congruences satisfied by the overpartition analogue, and some congruences satisfied by the associated smallest parts function.

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George E. Andrews, Atul Dixit, Daniel Schultz, Ae Ja Yee. 2016-03-14. Overpartitions related to the mock theta function $\omega(q)$. https://arxiv.org/abs/1603.04352

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