Pendants to the Euler Beta function

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.