  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.
