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Operational methods: An extension from ordinary monomials to multi-dimensional Hermite polynomials. (English) Zbl 1116.33014

The authors continue the investigation of the use of operational methods in deriving properties and formulas of ordinary and generalized special functions. In the present paper they show that the combination of monomiality principle with techniques of operational nature provides efficient tools which allow the derivation of several relations involving multi-dimensional Hermite polynomials.
In order to illustrate their approach the authors first derive some properties for the Kampé de Fériet polynomials defined by \[ H_{n}^{(3)}(x,\,y)=n!\sum_{r=0}^{[n/3]}\frac{x^{n-3r}\,y^{r}}{r!(n-3r)!}. \] They also prove various formulas for generalized Hermite polynomials \[ H_{n}^{(m)}(x,\,y)=n!\sum_{r=0}^{[n/m]}\frac{x^{n-mr}\,y^{r}}{r!(n-mr)!}, \] as well as the three variable Hermite polynomials \[ H_{n}^{(3,2)}(x,\,z,\,y)=n!\sum_{r=0}^{[n/3]}\frac{H_{n-3r}^{(2)}(x,\,z)\;y^{r}}{r!(n-3r)!}. \]

MSC:

33C47 Other special orthogonal polynomials and functions
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
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References:

[1] Appell P., Fonctions Hypergéometriques et Hypersphériques: Polinômes d’ Hermite (1926)
[2] DOI: 10.2307/1968431 · Zbl 0009.21202 · doi:10.2307/1968431
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