An integral representation for the product of parabolic cylinder functions

Abstract

This paper uses the convolution theorem of the Laplace transform to derive an inverse Laplace transform for the product of two parabolic cylinder functions in which the orders as well as the arguments differ. This result subsequently is used to obtain an integral representation for the product of two parabolic cylinder functions D ν (x)D μ (y). The integrand in the latter representation contains the Gaussian hypergeometric function or alternatively can be expressed in terms of the associated Legendre function of the first kind. ARTICLE HISTORY

An alternative integral representation for the product of two parabolic cylinder functions

“…The restrictions on x and y in both expressions are identical but the representations differ considerably in terms of the condition for convergence for the orders, namely Re(ν + μ) < 1 in [7] versus Re(ν) < 0 in Equation (3.1). This property of integral representation (3.1) has interesting consequences as will be noted below.…”

“…Note that the integrand in the integral representation (3.1) contains the parabolic cylinder function, whereas the representation for D ν (x)D μ (y) in Equation (3.1) in [7] was expressed in terms of the Gaussian hypergeometric function or the associated Legendre function of the first kind. The restrictions on x and y in both expressions are identical but the representations differ considerably in terms of the condition for convergence for the orders, namely Re(ν + μ) < 1 in [7] versus Re(ν) < 0 in Equation (3.1).…”

“…Subsequently, Nasri [6] derived integral representations for unrelated orders but identical or opposite arguments, D ν (±x)D ν+μ−1 (x). Recently, Veestraeten [7] used the convolution theorem of the Laplace transform to obtain an integral representation in which both the arguments and the orders are unrelated, i.e…”

Some Laplace transforms and integral representations for parabolic cylinder functions and error functions

“…Plugging the transform (27) into the latter expression and simplifying the result via the linear transformation formula (24) gives (28).…”

Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes

“…These allow the results of Ditlevsen [11] to be derived. Another useful result, related to the idea of analytic continuation, is the reflection formula, analogous to that of the Gamma function and derived in the same way, but not nearly as well-known, to the extent that it appears to be a new result (see Appendix) despite integral representations of products of parabolic cylinder functions remaining of interest in the special functions community [35,51,52]:…”