Humbert Polynomials and Functions in Terms of Hermite Polynomials Towards Applications to Wave Propagation

By starting from the standard definitions of the incomplete two-variable Hermite polynomials, we propose non-trivial generalizations and we show some applications to the Bessel-type functions as the Humbert functions. We also present a generalization of the Laguerre polynomials in the same context of the incomplete-type and we use these to obtain relevant operational techniques for the Humbert-type functions. Final considerations are inserted to include the problem of wave propagation in the present theoretical framework.