Combinatorial identities associated with a bivariate generating function for overpartition pairs (Q2104905)

\textit{S. H. Chan} and \textit{R. Mao} [J. Math. Anal. Appl. 394, No. 1, 408--415 (2012; Zbl 1294.11182)] established the following two \(q\)-series identities: \begin{align*} \sum_{n=0}^\infty \frac{(x,1/x;q)_nq^n}{(zq,q/z;q)_n} &= \frac{(1-z)^2}{(1-z/x)(1-xz)} + \frac{z(x,1/x;q)\infty}{(1-z/z)(1-xz)(zq,q/z;q)_\infty};\\ \sum_{n=0}^\infty \frac{(x,q/x;q)_nq^n}{(z,q/z;q)_n} & = \frac{1}{x(1-z/x)(1-q/(xz))} + \frac{(x,q/x;q)\infty}{z(1-x/z)(1-q/(xz))(z,q/z;q)_\infty}. \end{align*} In this paper, the authors obtain a three-parameter \(q\)-series identity that generalizes these identities of Chan and Mao: if \(\alpha,\gamma\in \mathbb{C}\) and \(\beta\in\mathbb{C}\setminus\{0,\alpha q,\gamma q,q^{-j},\alpha\gamma q^{j+2}|\ j\geq 0\}\), then \[ \sum_{n=0}^\infty \frac{(\alpha,\gamma;q)_nq^n}{(\beta,\alpha\gamma q^2/\beta;q)_n} = \frac{(1-q/\beta)(1-\alpha\gamma q\beta)}{(1-\gamma q/\beta)(1-\alpha q/\beta)} + \frac{\beta^{-1}q}{(1-\alpha q/\beta)(1-\gamma q/\beta)} \cdot \frac{(\alpha,\gamma;q)_\infty}{(\beta,\alpha\gamma q^2/\beta;q)_\infty}. \] New results of combinatorial significance in connection with a function counting certain overpartition pairs introduced by \textit{K. Bringmann} et al. [Int. Math. Res. Not. 2010, No. 2, 238--260 (2010; Zbl 1230.05034)] are provided in this context.