An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

Hermite polynomials on the plane

The space of bivariate generalised Hermite polynomials of degree n is invariant under rotations. We exploit this symmetry to construct an orthonormal basis for which consists of the rotations of a single polynomial through the angles , ℓ=0,...n. Thus we obtain an orthogonal expansion which retains as much of the symmetry of as is possible. Indeed we show that a continuous version of this orthogonal expansion exists.      

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

     

Characterizations of generalized Hermite and sieved ultraspherical polynomials

A new characterization of the generalized Hermite polyno-
mials and of the orthogonal polynomials with respect to the measure
is derived which is based on a ``reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.

     

On linearization and connection coefficients for generalized Hermite polynomials

We consider the problem of finding explicit formulas, recurrence relations and sign properties for both connection and linearization coefficients for generalized Hermite polynomials. Most of the computations are carried out by the computer algebra system Maple using appropriate algorithms.     

A class of generalized complex Hermite polynomials

A class of generalized complex polynomials of Hermite type, suggested by a special magnetic Schrödinger operator, is introduced and some related basic properties are discussed.     

Rotational symmetry of the Hermite projection operators

We calculate an integral formula for the Hermite projection operators. We give some applications of our formula. We also give a short proof of a recent theorem of Thangavelu.

     

Zero sets of bivariate Hermite polynomials

We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n+m$n+m$ consists of exactly n+m$n+m$ disjoint branches and possesses n+m$n+m$ asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R²${\mathbb{R}}^2$, are completely different for the three families analyzed.     

Operational rules and a generalized Hermite polynomials

In this paper, we use operational rules associated with three operators corresponding to a generalized Hermite polynomials introduced by Szegö to derive, as far as we know, new proofs of some known properties as well as new expansions formulae related to these polynomials.     

A note on Hermite polynomials of several variables

The present paper deals with an extension of certain results obtained by Burchnall for Hermite polynomials to similar results for Hermite polynomials of several variables.     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

Lie-theoretic generating relations of Hermite 2D polynomials

In this paper, we derive some generating relations involving Hermite 2D polynomials (H2DP) H_m,n(U;x,y), of two variables with an arbitrary 2D matrix U as parameter using Lie-theoretic approach. Certain (known or new) generating relations for the polynomials related to H2DP are also obtained as special cases.     

A new extension of Hermite matrix polynomials and its applications

In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Chebyshev matrix polynomials of the second kind are introduced.     

A curious q-analogue of Hermite polynomials

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.     

A globally uniform asymptotic expansion of the hermite polynomials

In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn（√2n＋1α）to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme （SIAM J.Math.Anal.,（1994）,25：304-321）.A new estimate for the remainder is given.     

Solutions of a system of forced burgers equation in terms of generalized laguerre polynomials

In this article, we obtain explicit solutions of a linear PDE subject to a class of radial square integrable functions with a monotonically increasing weight function |x|ⁿ⁻¹eβ|x|²/2,${\left|x\right|}^{n-1}{e}^{\beta {\left|x\right|}^2}/2,$, β≥0, x ∈?n$\beta \ge 0,\hspace{0.17em}x\hspace{0.17em}\in {?}^n$. This linear PDE is obtained from a system of forced Burgers equation via the Cole-Hopf transformation. For any spatial dimension n > 1, the solution is expressed in terms of a family of weighted generalized Laguerre polynomials. We also discuss the large time behaviour of the solution of the system of forced Burgers equation.     

Asymptotic Expansions of Hermite Functions on Compact Lie Groups

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let _t be the heat kernel on G; that is, _t is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from _t and its derivatives, are the compact group analogs of the classical Hermite polynomials on . Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on in the limit of small t, where the Hermite functions are viewed as functions on via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of . For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.     

Some families of hypergeometric polynomials and associated integral representations

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated single-, double-, and triple-integral representations. Some known or new consequences of the general results presented here, involving such classical orthogonal polynomials as the Jacobi, Laguerre, Hermite, and Bessel polynomials, and various other relatively less familiar hypergeometric polynomials, are also considered. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Volterra Integral Equation of Hermite Matrix Polynomials

The primary purpose of this paper is to present the Volterra integral equation of the two-variable Hermite matrix polynomials. Moreover, a new representation of these matrix polynomials are established here.