Generalization of Hermite polynomials
The generalized Hermite polynomials are defined. Let \(\Phi_ r({\mathbb{R}}^ m)\) be a space of symmetric r-linear forms on \({\mathbb{R}}^ m\). We denote by \({\mathcal F}_ s(n,r,\mu)\) the space of functions \(G_ r(x_ 1,...,x_ n;z_ 1,...,z_ r)\) acting from \(({\mathbb{R}}^ m)^ n\) to \(\Phi_ r({\mathbb{R}}^ m)\) with the following properties: i) they are symmetric with respect to \(x_ 1,...,x_ n\) and \(z_ 1,...,z_ r.\) ii) For any orthogonal \(U\in L({\mathbb{R}}^ m)\) \[ G_ r(Ux_ 1,...,Ux_ n;Uz_ 1,...,Uz_ r)=G_ r(x_ 1,...,x_ n;z_ 1,...,z_ r). \] iii) \(\int_{{\mathbb{R}}^ n}Sp Gr(x_ 1,...,x_ n;...)*C_ r(x_ 1,...,x_ n;...)\prod^{n}_{k=1}\Phi (x_ k)dx_ 1...dx_ n,\) where \(\Phi (x)=(2\pi)^{-(n/2)}\exp \{-(x,x)\}\) and \(\mu\) is a measure in \(({\mathbb{R}}^ m)^ n\) with the density \(\prod^{n}_{k=1}\Phi (x_ k)\). \({\mathcal F}_ s(n,r,\mu)\) is considered as a Hilbert space with respect to the scalar product \[ <G_ r,H_ r>=\int Sp G_ r(x_ 1,...,x_ n;...)*H_ r(x_ 1,...,x_ n;...)d\mu. \] The generalized Hermite polynomials are then constructed by the orthogonalization procedure. Some of their properties are investigated.
scientific article; zbMATH DE number 4200603