Explicit Formulas Associated with Some Families of Generalized Bernoulli and Euler Polynomials

In this paper, we propose and derive several new explicit formulas of the generalized Bernoulli and Euler polynomials in terms of the generalized Stirling numbers of the second kind. A study of some families of the modified generalized Euler polynomials yields an interesting algorithm for calculating the generalized Euler polynomials.     

A New Type of Euler Polynomials and Numbers

By defining two specific exponential generating functions, we introduce a kind of Euler polynomials and study its basic properties in detail. As an application of the introduced polynomials, we use them in computing some new series of Taylor type that contain the associated Euler numbers \(E_n(0)\) where \(E_n(x)\) is the Euler polynomial.     

A Note on a Generating Function for the Generalized Hermite Polynomials

Several generating-function relations involving the polynomials {ie1}, and their natural generalization {ie2}, are discussed. A hitherto seemingly unnoticed fact on the equivalence of certain known generating functions is also pointed out.     

Zeros of Generalized Hermite Polynomials and Ensemble of Gaussian Random Matrices

Mathematical Notes - The main purpose of the present paper is to investigate the global asymptotic eigenvalue density of the fixed-trace generalized Gaussian ensemble of random matrices. To answer...     

The Mehler Formula for the Generalized Clifford-Hermite Polynomials

The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler＇s formula for the generalized Clifford-Hermite polynomials of Clifford analysis.     

Some Generalized Clifford-Jacobi Polynomials and Associated Spheroidal Wavelets

In the present paper, by extending some fractional calculus to the framework of Clifford analysis, new classes of wavelet functions are presented. Firstly, some classes of monogenic polynomials are provided based on 2-parameters weight functions which extend the classical Jacobi ones in the context of Clifford analysis. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rules have been proved. The main tool reposes on the extension of fractional derivatives, fractional integrals and fractional Fourier transforms to Clifford analysis.     

高阶Bernoulli多项式和高阶Euler多项式的新计算公式

使用发生函数方法,利用两种第一类Stirling数给出高阶Bernoulli多项式和高阶Euler多项式的简捷计算公式.     

Euler Pseudoprime Polynomials and Strong Pseudoprime Polynomials

It is usual to emphasize the analogy between the integers and polynomials with coefficients in a finite field, comparing different notions in the two points of view. We introduce a particular rank one Drinfeld module to get an exponentiation for polynomials and then define the notions of Euler pseudoprimes and strong pseudoprimes for polynomials with coefficients in a finite field. As for the integers, we have Solovay–Strassen and Miller–Rabin tests for polynomials.     

Bernoulli多项式和Euler多项式的关系

本文给出了 Bernoulli- Euler数之间的关系和 Bernoulli- Euler多项式之间的关系 ,从而深化和补充了有关文献中的相关结果 .     

Asymptotic Distributions of Zeros of Quadratic Hermite–Pade Polynomials Associated with the Exponential Function

The asymptotic distributions of zeros of the quadratic Hermite--Pad\'{e} polynomials $p_{n},q_{n},r_{n}\in{\cal P}_{n}$ associated with the exponential function are studied for $n\rightarrow\infty$. The polynomials are defined by the relation $$(*)\qquad p_{n}(z)+q_{n}(z)e^{z}+r_{n}(z)e^{2z}=O(z^{3n+2})\qquad\mbox{as} \quad z\rightarrow0,$$ and they form the basis for quadratic Hermite--Pad\'{e} approximants to $e^{z}$. In order to achieve a differentiated picture of the asymptotic behavior of the zeros, the independent variable $z$ is rescaled in such a way that all zeros of the polynomials $p_{n},q_{n},r_{n}$ have finite cluster points as $n\rightarrow\infty$. The asymptotic relations, which are proved, have a precision that is high enough to distinguish the positions of individual zeros. In addition to the zeros of the polynomials $p_{n},q_{n},r_{n}$, also the zeros of the remainder term of (*) are studied. The investigations complement asymptotic results obtained in [17].     

Generalized Coherent States for the q-Oscillator Associated with Discrete q-Hermite Polynomials

We continue studying generalized coherent states of the Barut-Girardello type for oscillator-like systems related to a given set of orthogonal polynomials. In this paper we construct a family of coherent states associated with discrete q-Hermite polynomials of the II-type and prove the overcompleteness of this family by constructing the measure in the unity decomposition for this family of coherent states. Bibliography: 49 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 48–66.     

关于Hermite多项式的一个猜想的证明

王俊禹,孔令彬提出如下猜想:Hek(s)∑m 0(-1)mdmHedsk-mm(s)+Hek+1(s)m∑1(-1)mdm-1dHsme-k-1m(s)=k!,这里Hek(s)=(-1)kes2/2ddskk(e-s2/2),k=0,1,2,…,为Hermite多项式.我们给出这一猜想的证明.     

Hermite Subdivision Schemes and Taylor Polynomials

We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is C ⁰ and ℋ is C ^d . We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.      

Abscissa of Absolute Convergence of a Class of Generalized Euler Products

Mathematical Notes -     

Orthogonal Polynomials Associated with Related Measures and Sobolev Orthogonal Polynomials

Connection between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), are looked at. The results are applied to obtain information regarding Sobolev orthogonal polynomials associated with certain pairs of measures.     

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.     

Asymptotic Approximations between the Hahn-Type Polynomials and Hermite, Laguerre and Charlier Polynomials

It has been shown in Ferreira et al. (Adv. Appl. Math 31:61–85, [2003]), López and Temme (Methods Appl. Anal. 6:131–196, [1999]; J. Cpmput. Appl. Math. 133:623–633, [2001]) that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions. In this paper we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.      

A New Integral Involving the Product of Bessel Functions and Associated Laguerre Polynomials

A new result for integrals involving the product of Bessel functions and Associated Laguerre polynomials is obtained in terms of the hypergeometric function. Some special cases of the general integral lead to interesting finite and infinite series representations of hypergeometric functions.