More Infinite Products: Thue-Morse and the Gamma function

TL;DR

This paper explores infinite products involving the Thue-Morse sequence and rational functions, aiming to find new expressions in terms of known constants and extend the understanding of such products related to the Gamma function.

Contribution

It introduces new rational functions for which infinite products with the Thue-Morse sequence can be expressed in known constants, expanding the theory of such products.

Findings

Identified specific rational functions yielding known constant expressions

Extended the class of infinite products involving Thue-Morse sequence

Provided new links between infinite products and special constants

Abstract

Letting $(t_n)$ denote the Thue-Morse sequence with values $0, 1$, we note that the Woods-Robbins product $$ \prod_{n \geq 0} \left(\frac{2n+1}{2n+2}\right)^{(-1)^{t_n}} = 2^{-1/2} $$ involves a rational function in $n$ and the $\pm 1$ Thue-Morse sequence $((-1)^{t_n})_{n \geq 0}$. The purpose of this paper is twofold. On the one hand, we try to find other rational functions for which similar infinite products involving the $\pm 1$ Thue-Morse sequence have an expression in terms of known constants. On the other hand, we also try to find (possibly different) rational functions $R$ for which the infinite product $\prod R(n)^{t_n}$ also has an expression in terms of known constants.