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

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.

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

“…This expression is equivalent to Equation (2.1) in [5] that subsequently was shown to also yield the expressions for [3,4], respectively. Setting y = x in the integral representation (3.1), using the substitution u = 1 2 t and employing the identities sinh(2u) = (1 − exp(−4u))/2 exp(−2u) and coth(u) = (1 + exp(−2u))/(1 − exp(−2u)) gives…”

“…This property of integral representation (3.1) has interesting consequences as will be noted below. The integral representation (3.1) specializes into the expressions there were obtained in [3][4][5][6]. Setting μ = ν in Equation (3.1) gives…”

“…The resulting Nicholson-type integral representation for D ν (x)D μ (y) is first shown to specialize into the aforementioned expressions in [3][4][5][6] …”

“…In the 1930s, Meijer [1] and Bailey [2] (±x). Later, Glasser [4] and Veestraeten [5] obtained an integral expression for the product of two parabolic cylinder functions with identical orders but unrelated arguments, D ν (x)D ν (y). Subsequently, Nasri [6] derived integral representations for unrelated orders but identical or opposite arguments, D ν (±x)D ν+μ−1 (x).…”

An integral representation for the product of parabolic cylinder functions

“…Integral representations, on the other hand, for the case of unrelated orders but identical or opposite arguments, D ν (±x)D ν+μ−1 (x), were derived in [3]. The approach in [4] offered a first attempt at allowing simultaneously for differing arguments as well as orders. However, this approach still required the orders in each of the separate integral representations to be linearly related.…”