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Wallis type formula and a few versions of the number \(\pi\) in \(q\)-calculus. (English) Zbl 07822789

Silvestrov, Sergei (ed.) et al., Non-commutative and non-associative algebra and analysis structures. SPAS 2019. Selected papers based on the presentations at the international conference on stochastic processes and algebraic structures – from theory towards applications, Västerås, Sweden, September 30 – October 2, 2019. Cham: Springer. Springer Proc. Math. Stat. 426, 741-759 (2023).
Summary: In this paper, we expose a geometrical interpretation of the \(q\)-Wallis formula. We construct plane regions which consist of rectangles whose edges’ lengths are directly connected with factors in this formula. These regions are bounded by quarters of inside and outside circles from which we get estimates and conclusions about the number \(\pi_q\).
For the entire collection see [Zbl 07756361].

MSC:

05A30 \(q\)-calculus and related topics
11B65 Binomial coefficients; factorials; \(q\)-identities
11Y60 Evaluation of number-theoretic constants
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
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