Some double series for $\pi$ and their $q$-analogues

TL;DR

This paper derives new double series representations for pi and their q-analogues by applying partial derivatives to hypergeometric series formulas, expanding the mathematical understanding of pi's series expansions.

Contribution

It introduces novel double series for pi and their q-analogues using hypergeometric series derivatives, providing new formulas in the field.

Findings

New double series formulas for pi derived

q-analogues of these series established

Enhanced understanding of hypergeometric series applications

Abstract

By applying the partial derivative operator to several summation formulas for hypergeometric series, we prove several double series for $\pi$ in this paper. Similarly, we also establish several $q$-analogues of them.