Some integral representations and limits for (products of) the parabolic cylinder function

TL;DR

This paper derives new integral representations and limits for products of parabolic cylinder functions using inverse Laplace transforms and process limits, expanding the analytical tools for these special functions.

Contribution

It introduces novel integral representations and limit formulas for products of parabolic cylinder functions, leveraging inverse Laplace transforms and process limiting behavior.

Findings

Derived new integral representations for products of parabolic cylinder functions.

Calculated limits for these functions using Ornstein-Uhlenbeck and Brownian motion processes.

Extended the results to various order combinations via recurrence relations.

Abstract

Veestraeten [1] recently derived inverse Laplace transforms for Laplace transforms that contain products of two parabolic cylinder functions by exploiting the link between the parabolic cylinder function and the transition density and distribution functions of the Ornstein-Uhlenbeck process. This paper first uses these results to derive new integral representations for (products of two) parabolic cylinder functions. Second, as the Brownian motion process with drift is a limiting case of the Ornstein-Uhlenbeck process also limits can be calculated for the product of gamma functions and (products of) parabolic cylinder functions. The central results in both cases contain, in stylised form, D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-1}(y) such that the recurrence relation of the parabolic cylinder function straightforwardly allows to obtain integral representations and limits also for countless other combinations in the orders such as D_{v}(x)D_{v-3}(y) and D_{v+1}(x)D_{v}(y).