The construction of $q$-analogues via ${}_3{\varphi }_2$-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 ${}_3{\varphi }_2$ 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, Apéry, Chu, Chu and Zhang, Fabry, Guillera, Ramanujan, and Zeilberger.

Submitted for publication