Exploring the Extended Beta-Logarithmic Function: Matrix Arguments and Properties

Abstract

The beta-logarithmic function substantially generalizes the standard beta function, which is widely recognized for its significance in many applications. This article is devoted to the study of a generalization of the classical beta-logarithmic function in a matrix setting called the extended beta-logarithmic matrix function. The proofs of some essential properties of this extension, such as convergence, partial derivative formulas, functional relations, integral representations, inequalities, and finite and infinite sums, are established. Moreover, an application of the extended beta-logarithmic function in matrix arguments is proposed in probability theory. Further, numerical examples and graphical presentations of the new generalization are obtained.