$q$-Analogues of two Ramanujan-type formulas for $1/\pi$

TL;DR

This paper develops $q$-analogues of two classical Ramanujan formulas for 1/π, using $q$-WZ pairs, extending the understanding of these formulas in the context of basic hypergeometric series.

Contribution

It introduces new $q$-analogues of Ramanujan-type formulas for 1/π, leveraging $q$-WZ pairs, which was not previously established.

Findings

Derived explicit $q$-analogues of Ramanujan formulas for 1/π

Utilized $q$-WZ pairs from earlier work for proofs

Extended classical formulas into the $q$-series framework

Abstract

We give $q$-analogues of the following two Ramanujan-type formulas for $1/\pi$: \begin{align*} \sum_{k=0}^\infty (6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 4^k} =\frac{4}{\pi} \quad\text{and}\quad \sum_{k=0}^\infty (-1)^k(6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 8^k } =\frac{2\sqrt{2}}{\pi}. \end{align*} Our proof is based on two $q$-WZ pairs found by the first author in his earlier work.