  This paper studies a suitably normalized set of generalized Hermite
polynomials and sets down a relevant Mehler formula, Rodrigues formula, and
generalized translation operator. Weighted generalized Hermite polynomials are
the eigenfunctions of a generalized Fourier transform which satisfies an F. and
M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like
oscillator calculus, which generalizes the calculus associated with the quantum
mechanical simple harmonic oscillator, is studied in terms of these
polynomials.
