共查询到20条相似文献,搜索用时 15 毫秒
1.
Kung-Yu Chen H. M. Srivastava 《Proceedings of the American Mathematical Society》2005,133(11):3295-3302
In some recent investigations involving differential operators for generalized Laguerre polynomials, Herman Bavinck (1996) encountered and proved a certain summation formula for the classical Laguerre polynomials. The main object of this sequel to Bavinck's work is to prove a generalization of this summation formula for a class of hypergeometric polynomials. The demonstration, which is presented here in the general case, differs markedly from the earlier proof given for the known special case. The general summation formula is also applied to derive the corresponding result for the classical Jacobi polynomials.
2.
Kung-Yu Chen 《Journal of Mathematical Analysis and Applications》2004,298(2):411-417
In his recent investigations involving differential operators for some generalizations of the classical Laguerre polynomials, H. Bavinck [J. Phys. A Math. Gen. 29 (1996) L277-L279] encountered and proved a certain summation identity for the classical Laguerre polynomials. The main object of this sequel to Bavinck's work is to prove a generalization of this summation identity for the Srivastava-Singhal polynomials. The demonstration, which is presented here in the general case, differs markedly from the earlier proof given for the known special case. It is also indicated how the general summation identity can be applied to derive the corresponding result for one class of the Konhauser biorthogonal polynomials. 相似文献
3.
Using a general q-summation formula, we derive a generating function for the q-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the continuous q-Hahn polynomials. A new proof of the orthogonality relation for the big q-Jacobi polynomials is also given. A simple evaluation of the Nassrallah–Rahman integral is derived by using this summation formula. A new q-beta integral formula is established, which includes the Nassrallah–Rahman integral as a special case. The q-summation formula also allows us to recover several strange q-series identities. 相似文献
4.
This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summation formula is given. 相似文献
5.
We prove homological mirror symmetry for Lefschetz fibrations obtained as Sebastiani–Thom sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under Sebastiani–Thom summation of a polynomial of type D. 相似文献
6.
The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply
the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan’s
Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin
and Rota along with an extension of Foata’s bijection between Meixner endofunctions and bi-colored permutations, we find a
combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial
theorem that makes it possible to reduce the double sum to a single sum. 相似文献
7.
P. Heinrich 《Journal of Theoretical Probability》1996,9(4):1019-1027
Zero-one laws for polynomials in Gaussian random variables have already been studied.(7) They are established here by very simple arguments: Fubini's theorem and the rotational invariance of centered Gaussian measures. The proof is built on the Polarization formula that has received much attention in Refs. 1 and 5. Our point of view derives from the deep work of Borell.(2) In a natural way, these results extend to finite-order Gaussian chaos processes. 相似文献
8.
Masao Ishikawa 《The Ramanujan Journal》2008,16(2):211-234
In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions.
The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we
express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear
Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s
Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained
from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize
Stanley’s open problem.
相似文献
9.
In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8]. 相似文献
10.
We establish a functional central limit theorem for the empirical process of bivariate stationary long range dependent sequences
under Gaussian subordination conditions. The proof is based upon a convergence result for cross-products of Hermite polynomials
and a multivariate uniform reduction principle, as in Dehling and Taqqu [Ann. Statist. 17 (1989), 1767–1783] for the univariate
case. The effect of estimated parameters is also discussed. 相似文献
11.
We present here a further investigation for the classical Frobenius–Euler polynomials. By making use of the generating function methods and summation transform techniques, we establish some summation formulas for the products of an arbitrary number of the classical Frobenius–Euler polynomials. The results presented here are generalizations of the corresponding known formulas for the classical Bernoulli polynomials and the classical Euler polynomials. 相似文献
12.
Bruce C Berndt 《Journal of Number Theory》1975,7(4):413-445
Character versions of the Poisson and Euler-Maclaurin summation formulas are derived. Instead of Bernoulli numbers and polynomials which appear in the classical Euler-Maclaurin formula, generalized Bernoulli numbers and polynomials now appear. Many applications of the results are given. In particular, applications are made to the evaluation of L-functions, the derivation of a character analogue of the Lipschitz summation formula, and the examination of a new character version of the classical gamma function. 相似文献
13.
Yuan He 《The Ramanujan Journal》2017,43(2):447-464
In this paper, a further investigation for the Apostol–Bernoulli and Apostol–Euler polynomials is performed, and some summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials are established by applying some summation transform techniques. Some illustrative special cases as well as immediate consequences of the main results are also considered. 相似文献
14.
An infinite summation formula of Hall-Littlewood polynomials due to Kawanaka is generalized to a finite summation formula,
which implies, in particular, twelve more multiple q-identities of Rogers-Ramanujan type than those previously found by Stembridge and the last two authors. 相似文献
15.
In this paper, the author introduces the Legendre–Gould Hopper polynomials by combining the operational methods with the principle of monomiality. Generating functions, series definition, differential equation and certain other properties of Legendre–Gould Hopper polynomials are derived. Further, operational representations of these polynomials are established, which are used to get integral representations and expansion formulae. Certain summation formulae for these polynomials are also obtained. 相似文献
16.
By means of generating function and partial derivative methods, we investigate and establish several general summation formulas involving two classes of polynomials. The general results would apply to yield some identities for the Pell polynomials and Pell-Lucas polynomials, and other general polynomials can also be recovered in this paper. 相似文献
17.
Ghazala Yasmin 《分析论及其应用》2018,34(2)
In this article, the 2-variable general polynomials are taken as base with Peters polynomials to introduce a family of 2-variable Peters mixed type polynomials.These polynomials are framed within the context of monomiality principle and their properties are established. Certain summation formulae for these polynomials are also derived. Examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored. 相似文献
18.
Annie Cuyt Brahim BenouahmaneHamsapriye Irem Yaman 《Applied Numerical Mathematics》2011,61(8):929-945
It is well known that Gaussian cubature rules are related to multivariate orthogonal polynomials. The cubature rules found in the literature use common zeroes of some linearly independent set of products of basically univariate polynomials. We show how a new family of multivariate orthogonal polynomials, so-called spherical orthogonal polynomials, leads to symbolic-numeric Gaussian cubature rules in a very natural way. They can be used for the integration of multivariate functions that in addition may depend on a vector of parameters and they are exact for multivariate parameterized polynomials. Purely numeric Gaussian cubature rules for the exact integration of multivariate polynomials can also be obtained.We illustrate their use for the symbolic-numeric solution of the partial differential equations satisfied by the Appell function F2, which arises frequently in various physical and chemical applications. The advantage of a symbolic-numeric formula over a purely numeric one is that one obtains a continuous extension, in terms of the parameters, of the numeric solution. The number of symbolic-numeric nodes in our Gaussian cubature rules is minimal, namely m for the exact integration of a polynomial of homogeneous degree 2m−1.In Section 1 we describe how the symbolic-numeric rules are constructed, in any dimension and for any order. In Sections 2, 3 and 4 we explicit them on different domains and for different weight functions. An illustration of the new formulas is given in Section 5 and we show in Section 6 how numeric cubature rules can be derived for the exact integration of multivariate polynomials. From Section 7 it is clear that there is a connection between our symbolic-numeric cubature rules and numeric cubature formulae with a minimal (or small) number of nodes. 相似文献
19.
A. Melman 《Linear and Multilinear Algebra》2018,66(4):785-791
We propose an alternative proof of Pellet’s theorem for matrix polynomials that, unlike existing proofs, does not rely on Rouché’s theorem. A similar proof is provided for the generalization to matrix polynomials of a result by Cauchy that can be considered as a limit case of Pellet’s theorem. 相似文献
20.
Helmut Prodinger 《Proceedings Mathematical Sciences》2009,119(5):567-570
Using the explicit (Binet) formula for the Fibonacci polynomials, a summation formula for powers of Fibonacci polynomials
is derived straightforwardly, which generalizes a recent result for squares that appeared in Proc. Ind. Acad. Sci. (Math. Sci.) 118 (2008) 27–41. 相似文献