Pendants to the Euler Beta function

TL;DR

This paper introduces Cauchy siblings of the Euler Beta function, explores their properties, and identifies connections with classical means, highlighting the complexity of generalizing to multiple variables.

Contribution

It defines new Cauchy Beta functions inspired by the Euler Beta function and investigates their properties and relationships with classical means.

Findings

Some Cauchy Beta functions coincide with classical means

Closed formulas are difficult for multivariable cases

The Cauchy Beta functions belong to their class of Cauchy quotients (for Euler Beta)

Abstract

Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as the arithmetic or the logarithmic Cauchy one. Although the bivariable generalizations of Beta functions are obtained by elemantary integration, it seems difficult to obtain closed formulas for more than two variables. % The questions whether these Cauchy Beta functions belong to their respective class of Cauchy quotients is addressed and answered positively in the case of the Euler Beta function, but postponed to a future paper for all the other cases.