On a Mellin transform of the generalized Hermite polynomials (Q5931991)

For \(\mu >-1/2\), the generalized Hermite polynomials \(H^{(\mu)}_n(x)\) are the orthogonal polynomials corresponding to the weight function \(e^{-x^2}|x|^{2\mu}\) in \((-\infty,+\infty)\). They have been introduced by \textit{G. Szegő} [Orthogonal polynomials. Revised ed. (1959; Zbl 0089.27501)] as a generalization of the ordinary Hermite polynomials, which are recovered for \(\mu=0\). The paper under review computes the Mellin transform of \(H^{(\mu)}_n(ax) e^{-x}\) (where \(a\) is a parameter) as \(\Gamma(z) g_n(z;2a)\). The polynomials \(g_n(z;a)\) are explicitly given in terms of the hypergeometric function \({_2}{F_2}\) and are shown to satisfy a difference equation and a recurrence formula.