A note on products involving zeta(3) and Catalan's constant

TL;DR

This paper provides new proofs for product formulas involving zeta(3), pi, and Catalan's constant, connecting them to the Borwein-Dykshoorn function and its generalizations, thus advancing understanding of these special constants.

Contribution

It offers direct proofs for four product formulas involving special constants using generalized Borwein-Dykshoorn functions and relates these to the parameterized-Euler-constant function.

Findings

Provided proofs for two formulas using Borwein-Dykshoorn function

Derived proofs for two additional formulas via a generalization of the function

Expressed the Borwein-Dykshoorn function in terms of the parameterized-Euler-constant function

Abstract

In a recent paper Kachi and Tzermias give elementary proofs of four product formulas involving zeta(3), pi, and Catalan's constant. They indicate that they were not able to deduce these products directly from the values of a function introduced in 1993 by Borwein and Dykshoorn. We provide here such a proof for two of these formulas. We also give a direct proof for the other two formulas, by using a generalization of the Borwein-Dykshoorn function due to Adamchik. Finally we give an expression of the Borwein-Dykshoorn function in terms of the "parameterized-Euler-constant function" introduced by Xia in 2013, which happens to be a particular case of the "generalized Euler constant function" introduced by K. and T. Hessami Pilehrood in 2010.