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Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

In this paper we discuss generalized two-variable Chebyshev polynomials and their relevant relations; in particular, by using their integral representations, we prove some operational identities. The approach is based on the generalized two-variable Hermite polynomials and the integral representations of ordinary Chebyshev polynomials of first and second kind. In addition, we discuss how the families of generalized Chebyshev polynomials can be used to prove some interesting properties related to ordinary Chebyshev polynomials of first and second kind. A fundamental role, as we see, is played by the powerful operational techniques verified by the families of generalized Hermite polynomials.