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1.
The main object of this paper is to give analogous definitions of Apostol type (see [T.M. Apostol, On the Lerch Zeta function, Pacific J. Math. 1 (1951) 161-167] and [H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77-84]) for the so-called Apostol-Bernoulli numbers and polynomials of higher order. We establish their elementary properties, derive several explicit representations for them in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) Zeta function, and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical Bernoulli numbers and polynomials of higher order.  相似文献   

2.
Recently, Srivastava et al. introduced a new generalization of the Bernoulli, Euler and Genocchi polynomials (see [H.M. Srivastava, M. Garg, S. Choudhary, Russian J. Math. Phys. 17 (2010) 251-261] and [H.M. Srivastava, M. Garg, S. Choudhary, Taiwanese J. Math. 15 (2011) 283-305]). They established several interesting properties of these general polynomials, the generalized Hurwitz-Lerch zeta functions and also in series involving the familiar Gaussian hypergeometric function. By the same motivation of Srivastava’s et al. [11] and [12], we introduce and derive multiplication formula and some identities related to the generalized Bernoulli type polynomials of higher order associated with positive real parameters a, b and c. We also establish multiple alternating sums in terms of these polynomials. Moreover, by differentiating the generating function of these polynomials, we give a interpolation function of these polynomials.  相似文献   

3.
We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol-Bernoulli, Apostol-Euler polynomials and the polynomials involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.  相似文献   

4.
利用组合数学的方法,得到了一些包含高阶Genocchi数和广义Lucas多项式的恒等式,并且由此建立了Fibonacci数与Riemann Zeta函数的关系式.  相似文献   

5.
The aim of this paper is to study on the Genocchi polynomials of higher order on P, the algebra of polynomials in the single variable x over the field C of characteristic zero and P, the vector spaces of all linear functional on P. By using the action of a linear functional L on a polynomial p(x) Sheffer sequences and Appell sequences, we obtain some fundamental properties of the Genocchi polynomials. Furthermore, we give relations between, the first and second kind Stirling numbers, Euler polynomials of higher order and Genocchi polynomials of higher order.  相似文献   

6.
李桂贞 《大学数学》2006,22(4):100-103
讨论了高阶Genocchi数的性质,建立了一些包含高阶Genocchi数和高阶Euler-Bernoulli数的恒等式.  相似文献   

7.
Recently, Srivastava and Pintér proved addition theorems for the generalized Bernoulli and Euler polynomials. Luo and Srivastava obtained the anologous results for the generalized Apostol–Bernoulli polynomials and the generalized Apostol–Euler polynomials. Finally, Tremblay et al. gave analogues of the Srivastava–Pintér addition theorem for general family of Bernoulli polynomials. In this paper, we obtain Srivastava–Pintér type theorems for 2D‐Appell Polynomials. We also give the representation of 2D‐Appell Polynomials in terms of the Stirling numbers of the second kind and 1D‐Appell polynomials. Furthermore, we introduce the unified 2D‐Apostol polynomials. In particular, we obtain some relations between that family of polynomials and the generalized Hurwitz–Lerch zeta function as well as the Gauss hypergeometric function. Finally, we present some applications of Srivastava–Pintér type theorems for 2D‐Appell Polynomials. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

8.
It is well-known that the Euler polynomials E2n(x) with n 0 can be expressed as a polynomial Hn(x(x – 1)) of x(x – 1). We extend Hn(u) to formal power series for n < 0 and prove several properties of the coefficients appearing in these polynomials or series, which generalize some recent results, independently obtained by Hammersley [7] and Horadam [8], and answer a question of Kreweras [9]. We also deduce several continued fraction expansions for the generating function of Euler polynomials, some of these formulae had been published by Stieltjes [14] and by Rogers [12] without proof. These formulae generalize our earlier results concerning Genocchi numbers, Euler numbers and Springer numbers [5, 4].  相似文献   

9.
Hongmei Liu 《Discrete Mathematics》2009,309(10):3346-5728
In this paper, by the generating function method, we establish various identities concerning the (higher order) Bernoulli polynomials, the (higher order) Euler polynomials, the Genocchi polynomials and the degenerate higher order Bernoulli polynomials. Particularly, some of these identities are also related to the power sums and alternate power sums. It can be found that, many well known results, especially the multiplication theorems, and some symmetric identities demonstrated recently, are special cases of our results.  相似文献   

10.
高阶退化Bernoulli数和多项式   总被引:2,自引:0,他引:2  
刘国栋 《数学杂志》2005,25(3):283-288
本文研究了高阶退化Berrioulli数和多项式的两个显明公式,得到了一个包含高阶Bemoulli数和Stirling数的恒等式,并推广了F.H.Howard,S.Shirai和K.I.Sato的结果。  相似文献   

11.
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   

12.
高阶Bernoulli多项式和高阶Euler多项式的关系   总被引:7,自引:0,他引:7  
雒秋明  马韵新  祁锋 《数学杂志》2005,25(6):631-636
利用发生函数的方法,讨论了高阶Bernoulli数和高阶Euler数,高阶Bernoulli多项式和高阶Euler多项式之间的关系,得到了经典Bernoulli数和Euler数,经典Bernoulli多项式和Euler多项式之间的新型关系。  相似文献   

13.
This paper describes an approach to generalized Bernoulli polynomials in higher dimensions by using Clifford algebras. Due to the fact that the obtained Bernoulli polynomials are special hypercomplex holomorphic (monogenic) functions in the sense of Clifford Analysis, they have properties very similar to those of the classical polynomials. Hypercomplex Pascal and Bernoulli matrices are defined and studied, thereby generalizing results recently obtained by Zhang and Wang (Z. Zhang, J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math. 154 (11) (2006) 1622-1632).  相似文献   

14.
Two elegant representations are derived for the modified Chebyshev polynomials discussed by Witula and Slota [R. Witula, D. Slota, On modified Chebyshev polynomials, J. Math. Anal. Appl. 324 (2006) 321-343].  相似文献   

15.
我们得到Apostol-Bernoulli多项式的一个用Gauss超几何函数表示的新公式,并给出了它的一些特殊情况和应用.  相似文献   

16.
Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been investigated in the literature rather extensively. Recently, the present authors (Srivastava and Pinter in Appl Math Lett 17:375–380, 2004) obtained addition theorems and other relationships involving the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ , but also the generalized Genocchi polynomials ${G_n^{(\alpha)}(x)}$ , the Srivastava polynomials ${\mathcal{S}_{n}^{N}\left( x\right)}$ , several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.  相似文献   

17.
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   

18.
We prove a general symmetric identity involving the degenerate Bernoulli polynomials and sums of generalized falling factorials, which unifies several known identities for Bernoulli and degenerate Bernoulli numbers and polynomials. We use this identity to describe some combinatorial relations between these polynomials and generalized factorial sums. As further applications we derive several identities, recurrences, and congruences involving the Bernoulli numbers, degenerate Bernoulli numbers, generalized factorial sums, Stirling numbers of the first kind, Bernoulli numbers of higher order, and Bernoulli numbers of the second kind.  相似文献   

19.
In this paper we consider a family of generalized Fibonacci-type polynomials. These polynomials have a lot of similar properties to the generalized Jacobsthal-type polynomials. As an extension of the work of Djordjevi? [G.B. Djordjevi?, Mixed convolutions of the Jacobsthal type, Appl. Math. Comput. 186 (2007) 646-651], we give some recurrence relations and identities involving the generalized Fibonacci-type polynomials.  相似文献   

20.
Euler多项式的若干对称恒等式   总被引:1,自引:0,他引:1  
Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain some relations between the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers.  相似文献   

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