  We consider the Mellin transforms of certain generalized Hermite functions
based upon certain generalized Hermite polynomials, characterized by a
parameter $\mu>-1/2$. We show that the transforms have polynomial factors whose
zeros lie all on the critical line. The polynomials with zeros only on the
critical line are identified in terms of certain $_2F_1(2)$ hypergeometric
functions, being certain scaled and shifted Meixner-Pollaczek polynomials.
Other results of special function theory are presented.
