Polynomials of Meixnerʹs type in infinite dimensions—Jacobi fields and orthogonality measures

The classical polynomials of Meixnerʹs type—Hermite, Charlier, Laguerre, Meixner, and Meixner–Pollaczek polynomials—are distinguished through a special form of their generating function, which involves the Laplace transform of their orthogonality measure. In this paper, we study analogs of the latter three classes of polynomials in infinite dimensions. We fix as an underlying space a (non-compact) Riemannian manifold X and an intensity measure σ on it. We consider a Jacobi field in the extended Fock space over L2(X;σ), whose field operator at a point x∈X is of the form ∂x†+λ∂x†∂x+∂x+∂x†∂x∂x, where λ is a real parameter. Here, ∂x and ∂x† are, respectively, the annihilation and creation operators at the point x. We then realize the field operators as multiplication operators in L2(D′;μλ), where D′ is the dual of D≔C0∞(X), and μλ is the spectral measure of the Jacobi field. We show that μλ is a gamma measure for |λ|=2, a Pascal measure for |λ|>2, and a Meixner measure for |λ|<2. In all the cases, μλ is a Lévy noise measure. The isomorphism between the extended Fock space and L2(D′;μλ) is carried out by infinite-dimensional polynomials of Meixnerʹs type. We find the generating function of these polynomials and using it, we study the action of the operators ∂x and ∂x† in the functional realization.