Combinatorial identities associated with a bivariate generating function for overpartition pairs

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. Other applications include expressing a multi-sum involving N(r,s,m,n) in terms of the partition function p(n) and relating a certain double series to a weight 7/2 theta series.