New identities for the Glasser transform and their applications

TL;DR

This paper establishes new identities linking the Glasser transform with the $ ext{L}_2$-transform, deriving new Parseval-Goldstein type theorems that facilitate evaluation of integrals involving special functions.

Contribution

It introduces a novel iteration identity connecting the $ ext{L}_2$-transform and the Glasser transform, leading to new integral identities and evaluation techniques.

Findings

Derived a key iteration identity between $ ext{L}_2$-transform and Glasser transform.

Established new Parseval-Goldstein type theorems for these transforms.

Provided examples demonstrating applications to evaluate integrals of special functions.

Abstract

In the present paper the authors show that an iteration of the $\mathscr{L}_{2}$-transform by itself is a constant multiple of the Glasser transform. Using this iteration identity, a Parseval-Goldstein type theorem for $\mathscr{L}_{2}$-transform and the Glasser transform is given. By making use of these results a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here.