# Analytic study and statistical enforcement of extended beta functions imposed by Mittag-Leffler and Hurwitz-Lerch Zeta functions

## TL;DR

This paper introduces a new extended Beta function using advanced mathematical tools to improve analytical and statistical modeling capabilities.

## Contribution

The novel use of Mittag-Leffler and Hurwitz-Lerch Zeta functions as kernels to expand the Beta function's analytical and computational power.

## Key findings

- The extended Beta function was derived using Mittag-Leffler and Hurwitz-Lerch Zeta kernels.
- New derivative formulae and integral representations were obtained for enhanced mathematical modeling.
- The extended Beta function was applied to the Beta distribution, demonstrating its statistical significance.

## Abstract

Special Function Theory is used in many mathematical fields to model scientific progress, from theoretical to practical. This helps efficiently analyze the newly expanded Beta class of functions on a complicated domain. We use Mittag-Leffler and Hurwitz Lerch zeta (HLZ) kernels to produce the Beta function using the convolution tool. This special function advances a statistical implementation research approach. This unique function also discusses and gives analytical benefits, including functional and summation relations, Mellin transformations, and integral representations. Additionally, many derivative formulae are obtained. The statistical implementation of expanded Beta distribution using the suggested beta function was also conducted. We use the extended Beta function to create the new extended ordinary hypergeometric function and Kummer function. Derivative formulae, integral representations, generating functions, and fractional derivatives are also investigated.•Developed utilizing Mittag-Leffler and Hurwitz Lerch Zeta functions as kernels, delivering increased analytical and computational capabilities.•Comprises derivative formulae, integral representations, Mellin transformations, and generating functions, offering a comprehensive mathematical foundation.•Illustrates the use of the extended Beta function inside the Beta distribution, highlighting its statistical importance.

Developed utilizing Mittag-Leffler and Hurwitz Lerch Zeta functions as kernels, delivering increased analytical and computational capabilities.

Comprises derivative formulae, integral representations, Mellin transformations, and generating functions, offering a comprehensive mathematical foundation.

Illustrates the use of the extended Beta function inside the Beta distribution, highlighting its statistical importance.

Image, graphical abstract