An integral representation for the product of two parabolic cylinder functions having unrelated arguments

TL;DR

This paper derives an integral representation for the product of two parabolic cylinder functions with unrelated arguments, enabling new hyperbolic integral formulas and sum rules for these special functions.

Contribution

It introduces a novel integral representation for the product of parabolic cylinder functions with unrelated arguments, expanding analytical tools for these functions.

Findings

Provides an integral formula for $D_{0}(x)D_{0}(-y)$ with $Re extless0$ and $x extgreater y$

Derives hyperbolic integral identities from the representation

Establishes a sum rule relating these functions

Abstract

An integral representation is provided for the parabolic cylinder function product $D_{\mu}(x)D_{\mu}(-y)$ where $Re\,\mu<0$ and $x>y$ are unrelated. A few simple consequences are given in the form of hyperbolic integrals and a sum rule.