Abstract

In the present blog, we introduce several new integral transforms including the Ln-transform, the L2n-transform and P2n-transform generalizations of the classical Laplace transform and the classical Stieltjes transform as respectively. It is shown that the second iterate of the L2n-transform is essentially the P2n-transform. Using this relationship, a few new Parseval-Goldstein type identities are obtained. The theorem and the lemmas that are proven in this blog are new useful relations for evaluating infinite integrals of special functions. Some related illustrative examples are also given.

Introduction, definitions

Integral transforms are used in various branches of applied mathematics and among them, the Laplace transform is the most commonly used one in applications. Formulas for the classic Laplace transform, Goldstein's Parseval-type theorem for the classical Laplace transform, Yurekli's Parseval-Goldstein type theorem and Srivastava and Singh's Parseval-Goldstein type formula for the Widder potential transform, L2-transform, are given.\n

The Ln, L2n, and P2n Transforms

We now introduce a few generalized integral transforms. The Ln(n= 2k, k= 0,1,2, . . . , k∈N) transform is defined by the given formula and is a generalization of the Laplace transform. The L2n transform, a generalization of the L2-transform, and the Ln-transform are related to the Laplace transform and the L2-transform with the following formulas.

The Pn-transform and the P2n-transform

We further introduce the Pn-transform and the P2n-transform as follows. The identity relations between the Pn-transform, the Stieltjes-transform by the identity, Pn-transform and the Widder potential transform by means of the identity, P2n-transform and the Widder potential transform are also given.

Theorems and corollaries

In this section, Lemma 2.1 and Lemma 2.2 are presented proving the identities between the L2n-transform and the Ln-transform. The proofs are provided using definitions, changing the order of integration and using the definition of erfc(x) function.