  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. The central results in both cases contain, in
stylised form, D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-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 as D_{v}(x)D_{v-3}(y) and D_{v+1}(x)D_{v}(y).
