# Incomplete Riemann-Liouville fractional derivative operators and incomplete hypergeometric functions

## TL;DR

This paper introduces incomplete hypergeometric functions based on incomplete Pochhammer ratios and explores their properties, including integral representations and derivative formulas, along with defining an incomplete Riemann-Liouville fractional derivative operator.

## Contribution

It presents new incomplete hypergeometric functions and an associated fractional derivative operator, expanding the theoretical framework of special functions.

## Key findings

- Derived integral and transformation formulas for incomplete hypergeometric functions
- Established generating relations using the incomplete Riemann-Liouville fractional derivative
- Introduced new incomplete Pochhammer ratios linked to incomplete beta functions

## Abstract

In this paper, the incomplete Pochhammer ratios are defined in terms of the incomplete beta function $B_{y}(x,z)$. With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric and Appell's functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relation. Furthermore, an incomplete Riemann-Liouville fractional derivative operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.