Pré-Publication, Document De Travail Année : 2025

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

John Campbell

Résumé

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'' technique 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). We apply our $q$-WZ pairs to introduce and prove $q$-analogues of two of Ramanujan's series for $\frac{1}{\pi}$ and $q$-analogues of Guillera's first two series for $\frac{1}{\pi^2}$. We also apply our normalization method to further extend Guillera's WZ pairs by introducing hypergeometric expansions for $\frac{1}{\pi^2}$.

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hal-04973307 , version 1 (06-03-2025)

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John Campbell. $q$-analogues of $\pi$-formulas due to Ramanujan and Guillera. 2025. ⟨hal-04973307⟩
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