Some interesting explicit series representations, integrals and identities and their link to Jacobi,Laguerre and

Hermite polynomials are obtained.

An especially important case occurs for [alpha] = 2, [rho ] = 0, corresponding to the classical

Hermite polynomials. Note that the recurrence coefficients for general values of a and [rho ] are not known explicitly, but their asymptotic behavior has been established [14].

In Section 2 we introduce preliminary results about the n-polyanalytic function spaces and their relationship with the

Hermite polynomials. In Section 3 we prove that every Toeplitz operator with bounded horizontal symbol a(z) acting on Fock space is unitary equivalent to a multiplication operator [[gamma].sup.n,[alpha]] (x)I acting on ([L.sub.2]([R.sub.+])).sup.n, where [[gamma].sup.n,a] (x) is a continuous matrix-valued function on (-[infinity], [infinity]).

Polynomial chaos was first introduced by Wiener [6] where

Hermite polynomials were used to model stochastic processes with Gaussian random variables.

In this paper, we use the probabilistic orthonormal

Hermite polynomials for [[PSI].sub.i]([xi]) and, accordingly, f([xi]) = (1/[square root of 2[pi]])exp(-[[xi].sup.2]/2).

The wave functions in Schrodinger equation for the well-known potentials have been obtained on the orthogonal polynomials, such as Jacobi, generalized Laguerre, and

Hermite polynomials and the energy eigenvalues spectrum can be accessible for each case.

There are only a few explicit examples which include the formula for the exponential generating function of

Hermite polynomials [8] and the Glaisher-Crofton identity [9-11]

These facts, as well as cardinal basis functions perspective, motivated us in [11] to patch cubic

Hermite polynomials together to construct piecewise cubic fuzzy

Hermite polynomial and provide an explicit formula in a succinct algorithm to calculate the fuzzy interpolant in cubic case as a new replacement method for [4,10].

Finally, [P.sub.s]([[xi].sup.[alpha]]), [Q.sup.[beta].sub.s] ([[xi].sup.[alpha]]), [R.sup.[beta][gamma].sub.s] ([[xi].sup.[alpha]]), and [S.sup.123.sub.s] ([[xi].sup.[alpha]]) are suitable products of the

Hermite polynomials in (4).

Two equivalent versions of the

Hermite polynomials may be found in the literature: for n = 0, 1, 2...,

The Hermite analysis functions called [D.sub.n] [20, 21] are a set of orthogonal polynomials obtained from the product of a Gaussian window defined by g and the

Hermite polynomials [H.sub.n], where n in this case indicates the order of analysis.

Example 2

Hermite polynomials [{[H.sub.n](t)}.sub.n] forms an orthogonal polynomial system with respect to the linear functional L defined by [Lt.sup.k] = [EX.sup.k], where X is a Gaussian random variable.