Hilbert spaces of analytic functions and the Gegenbauer polynomials (Q1174570)

Let \(\mathbb{F}\) be the Fock type Hilbert space of analytic functions \(f(z)\) of \(n\) complex variables \(z=(z_ 1,z_ 2,\dots,z_ n)\in\mathbb{C}^ n\), with the scalar product \[ (f,g)=\pi^{-n}\int_{\mathbb{C}^ n}\overline{f(z)}g(z)\exp(-| z_ 1|^ 2-\cdots-| z_ n|^ 2)dz_ 1\cdots dz_ n, \] with \(dz_ 1\cdots dz_ n=dx_ 1\cdots dx_ ndy_ 1\cdots dy_ n\), \(z_ j=x_ j+iy_ j\), and let \(\mathbb{H}\) be the usual Hilbert space \(L^ 2(\mathbb{R}^ n)\). Bargmann constructed a unitary mapping \(A\) from \(\mathbb{H}\) to \(\mathbb{F}\) given by an integral operator whose kernel is related in some definite sense to the Hermite polynomials. More precisely, \(f=A\phi\) for \(\phi\in\mathbb{H}\) is defined by \[ f(z)=\int_{\mathbb{R}^ n}A(z,q)\phi(q)d^ nq \] with \[ A(z,q)=\pi^{-n/4}\prod_{j=1}^ n\exp\{-{1 \over 2}(z_ j^ 2+q_ j^ 2)+2^{1/2}z_ jq_ j\}. \] The purpose of the present paper is to show that similar constructions are possible for some other classical orthogonal polynomials.