Polynomials of Meixner's type in infinite dimensions-Jacobi fields and orthogonality measures

Abstract

The classical polynomials of Meixner's 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 $\sigma$ on it. We consider a Jacobi field in the extended Fock space over $L^2(X;\sigma)$, whose field operator at a point $x\in X$ is of the form $\di_x^\dag+\lambda\di_x^\dag \di_x+\di_x+\di^\dag_x\di_x\di_x$, where $\lambda$ is a real parameter. Here, $\di_x$ and $\di_x^\dag$ are, respectively, the annihilation and creation operators at the point $x$. We then realize the field operators as multiplication operators in $L^2({\cal D}';\mu_\lambda)$, where ${\cal D}'$ is the dual of ${\cal D}{:=}C_0^\infty(X)$, and $\mu_\lambda$ is the spectral measure of the Jacobi field. We show that $\mu_\lambda$ is a gamma measure for $|\lambda|=2$, a Pascal measure for $|\lambda|>2$, and a Meixner measure for $|\lambda|<2$. In all the cases, $\mu_\lambda$ is a L\'evy noise measure. The isomorphism between the extended Fock space and $L^2({\cal D}';\mu_\lambda)$ is carried out by infinite-dimensional polynomials of Meixner's type. We find the generating function of these polynomials and using it, we study the action of the operators $\di_x$ and $\di_x^\dag$ in the functional realization.