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

Recently, [Veestraeten D. An integral representation for the product of parabolic cylinder functions. Integral Transforms Spec Funct. 2017;28(1):15–21] derived an integral representation for Dν(x)Dμ(y) with Re(ν+μ)<1 that was expressed in terms of the Gaussian hypergeometric function. This paper obtains an alternative expression for Dν(x)Dμ(y) in which the integrand contains the parabolic cylinder function itself with the condition for convergence being at Re(ν)<0. The latter property is subsequently used to generate a new integral representation for Dμ(y) in which restrictions on the order μ are absent.