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.     

Some Linear Functionals Having Classical Orthogonal Polynomials as Moments

In this paper, we deal with some linear functionals on the vector space of polynomials whose moments are, in certain normalization, classical orthogonal polynomials (Hermite, Laguerre and Gegenbauer). We show that these linear functionals are semiclassical of class, at most, three. We give the coefficients in the three-term recurrence relations that the corresponding monic orthogonal polynomial sequences satisfy.     

Basis conversions among univariate polynomial representations

In this Note we provide a family of conversion algorithms relating Bernstein polynomials, monomials and the classical families of orthogonal polynomials, such as Jacobi, Gegenbauer, Legendre, Chebyshev, Laguerre and Hermite polynomials. To cite this article: R. Barrio, J.M. Peña, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Indefinite integrals of special functions from hybrid equations

ABSTRACT

Elementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.     

Parameter-based Fisher's information of orthogonal polynomials

The Fisher information of the classical orthogonal polynomials with respect to a parameter is introduced, its interest justified and its explicit expression for the Jacobi, Laguerre, Gegenbauer and Grosjean polynomials found.     

Characterization of q-Dunkl-classical symmetric orthogonal q-polynomials

The Ramanujan Journal - In this paper, we show that, up to a dilatation, the $$q^2$$ -analogue of generalized Hermite and $$q^2$$ -analogue of generalized Gegenbauer polynomials are the only...     

A new approach to the asymptotics of Sobolev type orthogonal polynomials

This paper deals with Mehler–Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points.     

The rodrigues type representations for a certain class of special functions

Summary An attempt is made here to present a systematic introduction to and several applications of a certain method of obtaining Rodrigues type representations for a fairly wide variety of sequences of special functions. The main results, contained in Theorems2 and3 below, are shown to apply not only to the Bessel polynomials, the classical orthogonal polynomials including, for instance, Hermite, Jacobi (and, of course, Gegenbauer, Legendre, and Tchebycheff), and Laguerre polynomials, and to their various generalizations studied in recent years, but also to such other special functions as the Bessel function and a certain class of generalized hypergeometric functions. Entrata in Redazione il 25 giugno 1977. This work was partially supported by the National Research Council of Canada under grants A-7353 and A-4027. For a preliminary report of this paper see Notices Amer. Math. Soc.,24 (1977), p. A-238, Abstract no. 77T-B43.     

Barut–Girardello Coherent States for the Gegenbauer Oscillator

We define the Gegenbauer oscillator and introduce a family of Barut–Girardello coherent states (eigenstates of a relevant annihilation operator) for this oscillator. Gegenbauer (ultraspherical) polynomials play the same role for this oscillator as Hermite polynomials do in the case of the usual boson oscillator. We establish the validity of unity decomposition for the introduced states and evaluate their overlap. These results reproduce similar results obtained earlier by the authors for Legendre and Chebyshev polynomials. Bibliography: 38 titles.     

Generalized polynomials and new families of generating functions

We show that the use of generalized polynomials, i.e. polynomials defined as discrete convolution of Hermite, Laguerre… polynomials can be exploited to explore new families of generating functions.

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Sunto Si dimostra che l'uso di polinomi generalizzati definiti come convoluzioni discrete di polinomi di Hermite, Laguerre etc. possono essere utilizzati per studiare nuove famiglie di funzioni generatrici.

     

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand‐alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15–16 digits) of the nodes and weights of the Gauss–Hermite and Gauss–Laguerre quadratures.     

The Combinatorics of Laguerre,Charlier, and Hermite Polynomials

This paper describes several combinatorial models for Laguerre, Charlier, and Hermite polynomials, and uses them to prove combinatorially some classical formulas. The so-called “Italian limit formula” (from Laguerre to Hermite), the Appel identity for Hermite polynomials, and the two Sheffer identities for Laguerre and Charlier polynomials are proved. We also give bijective proofs of the three-term recurrences. These three families form the bottom triangle in R. Askey's chart classifying hypergeometric orthogonal polynomials.     

Incomplete 2D Hermite polynomials: properties and applications

Incomplete forms of two-variable two-index Hermite polynomials are introduced. Their link with Laguerre polynomials is discussed and it is shown that they are a useful tool to study quantum mechanical harmonic oscillator entangled states. The possibility of developing the theory of complete 2D Hermite polynomials from the point of view of the incomplete forms is analyzed too. The orthogonality properties of the associated harmonic-oscillator functions are finally discussed.     

Monomiality principle,operational methods and family of Laguerre–Sheffer polynomials

In this paper, the Laguerre–Sheffer polynomials are introduced by using the monomiality principle formalism and operational methods. The generating function for the Laguerre–Sheffer polynomials is derived and a correspondence between these polynomials and the Sheffer polynomials is established. Further, differential equation, recurrence relations and other properties for the Laguerre–Sheffer polynomials are established. Some concluding remarks are also given.     

Coefficients of Wronskian Hermite polynomials

We study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the p-star.     

Bounds for extreme zeros of some classical orthogonal polynomials

We use mixed three term recurrence relations typically satisfied by classical orthogonal polynomials from sequences corresponding to different parameters to derive upper (lower) bounds for the smallest (largest) zeros of Jacobi, Laguerre and Gegenbauer polynomials.     

Application of a composition of generating functions for obtaining explicit formulas of polynomials

Using notions of composita and composition of generating functions, we obtain explicit formulas for the Chebyshev polynomials, the Legendre polynomials, the Gegenbauer polynomials, the Associated Laguerre polynomials, the Stirling polynomials, the Abel polynomials, the Bernoulli Polynomials of the Second Kind, the Generalized Bernoulli polynomials, the Euler Polynomials, the Peters polynomials, and the Narumi polynomials.     

Sharp estimates for the convergence rate of Fourier series in terms of orthogonal polynomials in L 2((a, b), p(x))

Sharp estimates are given for the convergence rate of Fourier series in terms of classical orthogonal polynomials in some classes of functions characterized by a generalized modulus of continuity in the space L ₂((a, b), p(x)). Expansions in terms of Laguerre, Hermite, and Jacobi polynomials are considered.     

Exponential operator splitting time integration for spectral methods

Pseudospectral spatial discretization by orthogonal polynomials and Strang splitting method for time integration are applied to second-order linear evolutionary PDEs. Before such a numerical integration can be used the original PDE is transformed into a suitable form. Trigonometric, Jacobi (and some of their special cases), generalized Laguerre and Hermite polynomials are considered. A double representation of a function (by coefficients of a polynomial expansion and by values at the nodes associated with a suitable quadrature formula) is used for numerical implementation so that it is possible to avoid calculations of matrix exponentials.     

Some characteristic property of the Meixner polynomials

A new characterization of the Meixner polynomials is established. It is based on the solution of a problem related to a previous result concerning the Laguerre polynomials. Solutions of analogous problems provide characteristic properties of the Laguerre and Hermite polynomials. These properties, which are derived from the two-variable polynomials, generalize, in turn, the previous ones.