The construction of $q$-analogues via $_3\phi_2$-series and $q$-difference equations

Abstract

We apply the EKHAD-normalization method given in our recent work to obtain, via the $q$-version of Zeilberger's algorithm, $q$-WZ pairs $(F, G)$ such that $\sum_{k = 0}^{\infty} F(0, k)$ may be expressed as a basic hypergeometric series of the form ${}_{3}\phi_2$ with multiple free parameters, and in such a way so that $\sum_{k=0}^{\infty} F(0, k) = \sum_{n=0}^{\infty} G(n, 0)$. In contrast to how previous applications of EKHAD-normalization relied on $q$-analogues for specific WZ pairs introduced by Guillera, our multiparameter approach provides a broad framework in the construction of $q$-analogues for accelerated series for universal constants such as $\pi$. We apply this multiparameter version of EKHAD-normalization to obtain and prove new $q$-analogues for accelerated hypergeometric series attributed to many authors, including (alphabetically) Adamchik and Wagon, Ap\'{e}ry, Chu, Chu and Zhang, Fabry, Guillera, Ramanujan, and Zeilberger.