Integral representations for products of two parabolic cylinder functions with different arguments and orders

TL;DR

This paper develops new integral formulas for products of two parabolic cylinder functions with different arguments and orders, enabling broader applications and connections to other special functions.

Contribution

It introduces novel integral representations for products of parabolic cylinder functions with different parameters and links them to hypergeometric and Bessel functions.

Findings

Derived integral representations for D_{nu}(x)D_{mu}(y) with x,y>0.

Obtained formulas for D_{nu}(-x)D_{mu}(y) using hypergeometric functions.

Specialized results for error functions and Bessel functions.

Abstract

This paper derives new integral representations for products of two parabolic cylinder functions. In particular, expressions are obtained for D_{nu}(x)D_{mu}(y), with x>0 and y>0, that allow for different orders and arguments in the two parabolic cylinder functions. Also, two integral representations are obtained for D_{nu}(-x)D_{mu}(y) by employing the connection between the parabolic cylinder function and the Kummer confluent hypergeometric function. The integral representations are specialized for products of two complementary error functions and of two modified Bessel functions of the second kind of order 1/4, as well as for the product of a parabolic cylinder function and a modified Bessel function of the first kind of order 1/4.