An alternative integral representation for the product of two parabolic cylinder functions. Issue 12 (2nd December 2017) - Veestraeten, Dirk
ABSTRACT: 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 forD ν ( x ) D μ ( y ) withRe ⁡ ( ν + μ ) < 1 that was expressed in terms of the Gaussian hypergeometric function. This paper obtains an alternative expression forD ν ( x ) D μ ( y ) in which the integrand contains the parabolic cylinder function itself with the condition for convergence being atRe ⁡ ( ν ) < 0 . The latter property is subsequently used to generate a new integral representation forD μ ( y ) in which restrictions on the order μ are absent.