Product of parabolic cylinder functions involving Laplace transforms of confluent hypergeometric functions

TL;DR

This paper derives integral representations and series expansions for products of parabolic cylinder functions using Laplace and Fourier transforms of confluent hypergeometric functions, providing new analytical tools.

Contribution

It introduces novel integral and series representations for products of parabolic cylinder functions via Laplace and Fourier transforms of confluent hypergeometric functions.

Findings

Derived integral representations for product of parabolic cylinder functions.

Transformed integral forms into Nicholson-type integrals.

Produced new series expansions for these products.

Abstract

In this paper, the product of parabolic cylinder functions $D_{\nu}(\pm z)D_{\nu+\mu-1}(z)$, with different parameters $\mu$ and $\nu$, are established in terms of Laplace and Fourier transforms of Kummer's confluent hypergeometric functions. The provided integral representations are transformed to easily yield Nicholson-type integral forms and used to derive other series expansions for products of parabolic cylinder functions.