Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by exp(−x2/2), which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a com...

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 ...

It is known that the Wigner distribution function in quantum Boltzmann equation not positive elsewhere, which causes trouble for its application to mesoscopic transport problem. Similarly, if we extend spinor equation, includes spin freedom usual distribution, and matrix therein Hermite, must make this non-Hermite Hermitization. So paper, propose a with Hermite then obtain equations of continui...

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 present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are wel...

In this paper we examine the zero and first order eigenvalue fluctuations for the β-Hermite and β-Laguerre ensembles, using the matrix models we described in [5], in the limit as β → ∞. We find that the fluctuations are described by Gaussians of variance O(1/β), centered at the roots of a corresponding Hermite (Laguerre) polynomial. We also show that the approximation is very good, even for sma...

We discuss the modeling of count data whose empirical distribution is both multi-modal and overdispersed, and propose the Hermite distribution with covariates introduced through the conditional mean. The model is readily estimated by maximum likelihood, and nests the Poisson model as a special case. The Hermite regression model is applied to data for the number of banking and currency crises in...

In this paper, we present Hermite-Hadamard inequality for p-convex functions in fractional integral forms. we obtain an integral equality and some Hermite-Hadamard type integral inequalities for p-convex functions in fractional integral forms. We give some HermiteHadamard type inequalities for convex, harmonically convex and p-convex functions. Some results presented in this paper for p-convex ...

In this paper, we introduce a new class of degenerate Hermite poly-Bernoulli polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of d...