A Comprehensive Study of the Extended Beta Function and Its Role in Special Functions

Abstract

The Beta function, defined traditionally as an integral involving two parameters, plays a significant role in many areas of mathematics, including analysis, number theory, and physics. However, the need for more general forms of the Beta function arises in a variety of advanced mathematical contexts. This paper aims to provide a thorough investigation into the extensions of the Beta function, focusing on its generalizations, key properties, and their implications in different branches of mathematics.

We begin by reviewing the classical Beta function, highlighting its relationship with the Gamma function, and explore its applications in integral calculus and probability theory. The study then moves to recent developments in the extension of the Beta function, which include higher-dimensional generalizations, the application of fractional orders, and extensions to complex and hypergeometric parameters. These extended forms are of particular interest in areas such as asymptotic analysis, quantum field theory, and complex analysis.