TL;DR
This paper revisits T. Rivoal's 2005 formula for 4/pi expressed as an infinite product, providing a new proof using a lemma from a 1988 paper and extending it to formulas involving digit blocks in base-B expansions.
Contribution
It offers a new proof of Rivoal's formula and generalizes it to formulas involving digit blocks in various base expansions.
Findings
New proof of Rivoal's infinite product formula for 4/π.
Extension of formulas to digit block occurrences in base-B expansions.
Connection between digit patterns and infinite product representations.
Abstract
In an unpublished 2005 paper T. Rivoal proved a formula giving 4/pi as the infinite product of factors (1 + 1/(k+1)) to a power involving the integer part of the logarithm of k in base 2 and a 4-periodic sequence. We show how a lemma in a 1988 paper of J. Shallit and the author allows us to prove that formula, as well as a family of similar formulas involving occurrences of blocks of digits in the base-B expansion of the integer k, where B is an integer larger than 1.
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Taxonomy
Topicssemigroups and automata theory · Mathematical Dynamics and Fractals · Coding theory and cryptography