An Infinite Family of Apéry-like Recurrences for Euler's Constant
Rotem Kalisch, Tali Monderer, Elyasheev Leibtag, Shachar Weinbaum, Michael Shalyt, Ido Kaminer
Abstract
Euler's constant γ, also known as the Euler-Mascheroni constant, remains one of the most elusive constants in mathematics. Despite extensive study, its irrationality is still unproven, and Apéry-like approximations of it are remarkably scarce. We present a novel identity that expresses γ in terms of a Meijer G-function and its first few derivatives. This identity is then generalized and used to systematically generate infinitely many Apéry-like approximations of γ using a Conservative Matrix Field construction based on the Meijer G-function.
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