The standard block orthogonal (SBO) polynomials Pi;n(x), 0 ≤ i ≤ n are real polynomials of degree n which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than i. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these po...

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are g...

In this paper we extend some recent results on moment identities, Hermite polynomials, and measure invariance properties on the Wiener space, to the setting of path spaces over Lie groups. In particular we prove the measure invariance of transformations having a quasi-nilpotent covariant derivative via a Girsanov identity and an explicit formula for the expectation of Hermite polynomials in the...

Polynomial expansions are of widespread use in a gas kinetic theory. For linearized Boltzmann equation, such an expansion is a basis of the wide-known Enskog-Chapman method [1], [2]. For the nonlinear case, this method was developed in Burnett's [3] and Grad's [4] works. As it was proved by Kumar [5], the best time-consuming expansion is that over so called spherical Hermite polynomials. Along ...

Using Casorati determinants of Charlier polynomials (ca n )n , we construct for each finite set F of positive integers a sequence of polynomials cF n , n ∈ σF , which are eigenfunctions of a second order difference operator, where σF is certain infinite set of nonnegative integers, σF ( N. For suitable finite sets F (we call them admissible sets), we prove that the polynomials cF n , n ∈ σF , a...

For use in calculating higher-order coherentand squeezedstate quantities, we derive generalized generating functions for the Hermite polynomials. They are given by ∑∞ n=0 z Hjn+k(x)/(jn + k)!, for arbitrary integers j ≥ 1 and k ≥ 0. Along the way, the sums with the Hermite polynomials replaced by unity are also obtained. We also evaluate the action of the operators exp[a(d/dx)] on well-behaved ...

In the present paper, we solve numerically Volterra integral equations of second kind, by the well known Galerkin method. For this, we derive a simple and efficient matrix formulation using Hermite polynomials as trial functions. Numerical examples are considered to verify the effectiveness of the proposed derivations and numerical solutions are compared with the existing methods available in t...

In this paper, we first define the multiple Appell polynomials and find several equivalent conditions for this class of polynomials. Then we give a characterization theorem that if multiple Appell polynomials are also multiple orthogonal, then they are the multiple Hermite polynomials.

The equilibrium positions of the multi-particle classical Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials associated with the classical root systems are described by the classical orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The eigenfunctions of the corresponding single-particle quantum CSM systems are also expressed in terms of the same ...

In this paper we give new proofs of some elementary properties of the Hermite and Laguerre orthogonal polynomials. We establish Rodriguestype formulae and other properties of these special functions, using suitable operators defined on the Lie algebra of endomorphisms to the vector space of infinitely many differentiable functions.