On a formula of the $q$-series ${}_{2k+4}{\phi }_{2k+3}$ and its applications

In this paper we apply a formula of the very-well poised ${}_{2k+4}{\phi }_{2k+3}$ to write a $k$-tuple sum of $q$-series as a linear combination of terms wherein each term is a product of expressions of the form $\frac{1}{(qy,q{y}^{-1};q{)}_{\mathrm{\infty }}}$. As an application, we shall express a variety of sums and double sums of $q$-series as linear combinations of infinite products. Our formulas are motivated by their connection to overpartition pairs.