$q$-analogues of $\pi$-formulas due to Ramanujan and Guillera

Abstract

The first known $q$-analogues for any of the $17$ formulas for $\frac{1}{\pi}$ due to Ramanujan were introduced in 2018 by Guo and Liu (J. Difference Equ. Appl. 29:505-513, 2018), via the $q$-Wilf-Zeilberger method. Through a "normalization" method, which we refer to as EKHAD-normalization, based on the $q$-polynomial coefficients involved in first-order difference equations obtained from the $q$-version of Zeilberger's algorithm, we introduce $q$-WZ pairs that extend WZ pairs introduced by Guillera (Adv. in Appl. Math. 29:599-603, 2002) (Ramanujan J. 11:41-48, 2006). We apply our EKHAD-normalization method to prove four new $q$-analogues for three of Ramanujan's formulas for $\frac{1}{\pi}$ along with $q$-analogues of Guillera's first two series for $\frac{1}{\pi^2}$. Our normalization method does not seem to have been previously considered in any equivalent way in relation to $q$-series, and this is substantiated through our survey on previously known $q$-analogues of Ramanujan-type series for $\frac{1}{\pi}$ and of Guillera's series for $\frac{1}{\pi^2}$. We conclude by showing how our method can be adapted to further extend Guillera's WZ pairs by introducing hypergeometric expansions for $\frac{1}{\pi^2}$.