Note on extensions of the beta function

TL;DR

This paper investigates the convergence properties of various extensions of the classical beta function, introduces a modified extension, and explores their numerical, geometric, and integral characteristics.

Contribution

It provides new insights into the convergence of beta function extensions and proposes a modified version with detailed properties and interpretations.

Findings

Convergence criteria for beta function extensions established

A new modified extension of the beta function introduced

Integral representations and geometric interpretations provided

Abstract

The classical beta function B(x; y) is one of the most fundamental special functions, due to its important role in various fields in the mathematical, physical, engineering and statistical sciences. Useful extensions of the classical Beta function has been considered by many authors. In the present paper, our main objective is to study the convergence of extensions of classical beta function and introduce modified extension of classical beta function. It is interpreted numerically and geometrically in the view of convergence, further properties and integral presentations are established.