Mellin Convolutions of Products and Ratios

TL;DR

This paper explores Mellin convolutions of products and ratios within the pathway family of functions, examining integral, G and H-function representations, and providing computable series forms for these convolutions.

Contribution

It introduces new integral and series representations for Mellin convolutions of pathway functions, expanding analytical tools for these convolutions.

Findings

Derived new integral representations for Mellin convolutions.

Established equivalences with G and H-functions.

Provided series expansions for computational purposes.

Abstract

Usually, convolution refers to Laplace convolution in the literature. But Mellin convolutions can yield very ueeful results. This aspect is illustrated in the coming sections. This paper deals with Mellin convolutions of products and ratios. Functions belonging to the pathway family of functions are considered. Several types of integral representations, their equivalent representations in terms of G and H-functions and their equivalent computable series representations are examined in this paper.