共查询到20条相似文献,搜索用时 15 毫秒
1.
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. 相似文献
2.
《分析论及其应用》2017,33(4):355-365
In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫_(-∞)~(+∞)(x~2)~mexp(-x~2){Hr (x)}2dx, ∫_0~∞exp(-x~2)H_(2k)(x)H_(2s+1)(x)dx,∫_0~∞exp(-x~2)H_(2k)(x)H_(2s)(x)dx and ∫_0~∞exp(-x~2)H_(2k+1)(x)H_(2s+1)(x)dx,are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of special functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite's polynomials by suitable simplifications of arbitrary parameters. 相似文献
3.
Yilmaz Simsek 《Mathematical Methods in the Applied Sciences》2019,42(18):7030-7046
The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed. 相似文献
4.
Geraldo Botelho 《Mathematische Nachrichten》2000,211(1):25-36
This paper introduces the class of almost summing homogeneous polynomials between Banach spaces. A characterization by means of vector‐valued sequences is proved, connections with the theory of absolutely summing polynomials are established and several examples are given. Some composition and coincidence theorems are obtained and the case of spaces with type or cotype is also investigated. 相似文献
5.
The Al–Salam & Carlitz polynomials are q–generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the q–analogue of the type–A Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with respect to a certain Jackson integral inner product. This inner product is normalized by deriving a q–analogue of the Mehta integral, and the corresponding normalization of the multivariable Al–Salam & Carlitz polynomials is derived from a Pieri–type formula. Various other special properties of the polynomials are also presented, including their relationship to the shifted Macdonald polynomials and the big–q Jacobi polynomials. 相似文献
6.
Predrag Rajković Franz Hinterleitner Sladjana Marinković 《Mathematical Methods in the Applied Sciences》2016,39(9):2358-2367
We have found the motivation for this paper in the research of a quantized closed Friedmann cosmological model. There, the second‐order linear ordinary differential equation emerges as a wave equation for the physical state functions. Studying the polynomial solutions of this equation, we define a new functional product in the space of real polynomials. This product includes the indexed weight functions which depend on the degrees of participating polynomials. Although it does not have all of the properties of an inner product, a unique sequence of polynomials can be associated with it by an additional condition. In the special case presented here, we consider the Hermite‐type weight functions and prove that the associated polynomial sequence can be expressed in the closed form via the Hermite polynomials. Also, we find their Rodrigues‐type formula and a four‐term recurrence relation. In contrast to the zeros of Hermite polynomials, which are symmetrically located with respect to the origin, the zeros of the new polynomial sequence are all positive. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
7.
8.
The multivariable BC type Askey-Wilson polynomials are considered for a parameter domain such that the orthogonality measure has partly discrete and partly continuous support. 相似文献
9.
Antonio J. Duran 《Journal of Approximation Theory》1999,100(2):2239
Ratio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind. 相似文献
10.
Bernoulli多项式和Euler多项式的关系 总被引:20,自引:1,他引:20
雒秋明 《数学的实践与认识》2003,33(3):119-122
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 . 相似文献
11.
通过讨论一类函数的高阶导数 ,建立了一些包含 Hermite-Laguerre多项式的恒等式 ,推广了著名的 Cauchy-Sheehan组合恒等式 . 相似文献
12.
Zhong-xuan LuoRen-hong Wang 《Journal of Mathematical Analysis and Applications》2002,268(1):171-183
In this paper, some important properties of orthogonal polynomials of two variables are investigated. The concepts of invariant factor for orthogonal polynomials of two variables are introduced. The presented results include Stieltjies type theorems for multivariate orthogonal polynomials and the corresponding asymptotic expansion formulas. 相似文献
13.
We introduce operators of q-fractional integration through inverses of the Askey–Wilson operator and use them to introduce a q-fractional calculus. We establish the semigroup property for fractional integrals and fractional derivatives. We study properties of the kernel of q-fractional integral and show how they give rise to a q-analogue of Bernoulli polynomials, which are now polynomials of two variables, x and y. As q→1 the polynomials become polynomials in x−y, a convolution kernel in one variable. We also evaluate explicitly a related kernel of a right inverse of the Askey–Wilson operator on an L2 space weighted by the weight function of the Askey–Wilson polynomials. 相似文献
14.
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.
相似文献
15.
Franck Beaucoup 《Journal of Approximation Theory》1998,93(3):434-440
We prove that any polynomial having all its roots in a closed half-plane, whose boundary contains the origin, has either one or two maximal points, and only one if it has at least one root in the open half-plane. This result concerns stable polynomials as well as polynomials having only real roots, including real orthogonal polynomials. 相似文献
16.
Gabriel Bengochea 《代数通讯》2013,41(7):3086-3095
We prove that the cofinite dual of the Hopf algebra of polynomials in several variables can be represented as a Hopf algebra ? of exponential polynomials that contains the polynomials as a Hopf subalgebra. We also present some algebras isomorphic to ? whose elements are rational functions or multi-sequences. 相似文献
17.
本文讨论了关于以下内积的正交多项式 :〈p(x) ,r(x)〉( u0 ,u(α,β) ) =∑∞k=0p(qk) r(qk) (qk-c) ak(b) k(q) k +λ∑∞k=0(Dqp) (qk) (Dqr) (qk) (aq) k(bq) k(q) k给出了它的一些代数性质以及和小 q-Jacobi多项式的关系 ,得到了在 C\([0 ,1 ]∪ H )的紧子集上Qn(x)P(α- 1,β- 1)n (x) n和 Pn(x)P(α- 1,β- 1)n (x) n的相对渐近性质 .其中 Qn(x)是 n次的小 q -Jacobi-Sobolev正交多项式 ,P(α- 1,β- 1)n (x)和 Pn(x)分别是关于线性泛函 u(α- 1,β- 1)和 u0 的首一的 n次正交多项式 . 相似文献
18.
19.
Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev
inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt
(J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain
interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt,
when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair.
This research is supported by grants from CNPq and FAPESP. 相似文献
20.
Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph. 相似文献