On some generating functions for the extended generalized Hermite polynomials

In a recent paper [G.C. Greubel, On some extensions and generating functions for the generalized Hermite polynomials associated with functions of parabolic cylinder (to be submitted)] the generalized Hermite polynomials of Wünsche were extended. In this work we derive several generating functions for the extended Hermite polynomials in terms of generalized Lauricella functions. In part I we derive several generating functions. In part II we derive an explicit relation for the extended generalized Hermite polynomials in terms of the Lauricella function.