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On formulas for π experimentally conjectured by Jauregui-Tsallis

On formulas for π experimentally conjectured by Jauregui-Tsallis
Tewodros Amdeberhan, David Borwein, Jonathan M. Borwein, Armin Straub — Journal of Mathematical Physics — Volume 53, Number 7, 2012, Pages 073708:1-15

Abstract

In a recent study of representing Dirac's delta distribution using \(q\)-exponentials, M. Jauregui and C. Tsallis experimentally discovered formulae for \(\pi\) as hypergeometric series as well as certain integrals. Herein, we offer rigorous proofs of these identities using various methods and our primary intent is to lay down an illustration of the many technical underpinnings of such evaluations. This includes an explicit discussion of creative telescoping and Carlson's Theorem. We also generalize the Jauregui--Tsallis identities to integrals involving Chebyshev polynomials. In our pursuit, we provide an interesting tour through various topics from classical analysis to the theory of special functions.

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BibTeX

@article{qpi-2012,
    author = {Tewodros Amdeberhan and David Borwein and Jonathan M. Borwein and Armin Straub},
    title = {On formulas for $\pi$ experimentally conjectured by {J}auregui--{T}sallis},
    journal = {Journal of Mathematical Physics},
    year = {2012},
    volume = {53},
    number = {7},
    pages = {073708:1--15},
    doi = {10.1063/1.4735283},
}