Multi-dimensional Chebyshev polynomials: a non-conventional approach

Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of differential equations, in particular in some special cases of Sturm-Liouville differential 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.