Proof of experimentally conjectured formulas for pi
This is a preprint by Tewodros Amdeberhan, David Borwein, Jonathan M. Borwein and Armin Straub.
Abstract
A recent paper by M. Jauregui and C. Tsallis contains a list of experimentally discovered formulae for π as finite series of ratios involving the gamma function. Herein, we offer rigorous proofs of these identities using various methods. We also prove and extend a corresponding integral identity conjectured in [M. Jauregui and C. Tsallis, New representations of π and Dirac delta using the nonextensive-statistical-mechanics q-exponential function, 2010]. In doing so, we pave way to an interesting tour through various topics from classical analysis and the theory of special functions.
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