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

Recently, [Veestraeten D. On the inverse transform of Laplace transforms that contain (products of) the parabolic cylinder function. Integr Transf Spec F 2015;26:859-871] 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 stylized form, Dv(x)Dv(y) and Dv(x)Dv−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 asDv(x)Dv−3(y) and Dv+1(x)Dv(y).