On incomplete exponential $\;_{r}R_{s}(P,Q,z)$ matrix function

TL;DR

This paper explores the properties, integral representations, and relations of incomplete exponential matrix functions, extending their theoretical framework and providing new formulas for their manipulation.

Contribution

It introduces generalized incomplete exponential matrix functions with new integral representations, addition, multiplication formulas, and recurrence relations.

Findings

Derived integral representations for generalized incomplete exponential matrix functions

Established addition and multiplication formulas for these functions

Presented recurrence matrix relations and summation formulas

Abstract

The recurrence matrix relations, differentiation formulas, and analytical and fractional integral properties of incomplete gamma matrix functions $\gamma(Q, x)$ and $\Gamma(Q, x)$ are all covered in this article. The generalized incomplete exponential matrix functions with their integral representations functions have been examined, along with some relevant characteristics of these functions such as integral representations functions . Additionally, the infinite summation relations and formulas for two sequences are shown, along with the generalized incomplete exponential matrix functions with the integral representation, addition formula for addition of two arguments, multiplication formula for multiplication of two arguments, and recurrence matrix relation.