Combinatorial identities associated with a bivariate generating function for overpartition pairs

TL;DR

This paper introduces a new three-parameter q-series identity that generalizes previous results, leading to novel combinatorial formulas and connections with classical functions like Chebyshev polynomials and partition functions.

Contribution

It derives a new three-parameter q-series identity that unifies and extends existing results, providing new combinatorial interpretations and closed-form evaluations related to overpartition pairs.

Findings

Provides a closed-form evaluation of a double series using Chebyshev polynomials.

Expresses a multi-sum involving overpartition pairs in terms of the partition function p(n).

Relates a double series to a weight 7/2 theta series via Shimura's result.

Abstract

We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving $N(r, s, m, n)$ in terms of just the partition function $p(n)$. Using a result of Shimura we also relate a certain double series with a weight 7/2 theta series.