共查询到20条相似文献,搜索用时 0 毫秒
1.
Takashi Agoh 《Journal of Number Theory》2009,129(8):1837-1847
Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as n(B0+B0)=−nBn−1−(n−1)Bn, is extended to n(Bk1+?+Bkm) for m?2 and arbitrary fixed integers k1,…,km?0. In the general case we prove an existence theorem for Euler-type formulas, and for m=3 we obtain explicit expressions. This extends the authors' previous work for m=2. 相似文献
2.
We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then
- $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B n . Among others, we obtain
- $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.
3.
The concept of a convolution identity for tensors is introduced and it is proved that any convolution identity for tensors on a finite-dimensional space follows from a convolution identity equivalent to the classical Cayley-Hamilton identity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 211–214, 1982. 相似文献
4.
Hans J.H. Tuenter 《Journal of Number Theory》2006,117(2):376-386
In the Frobenius problem with two variables, one is given two positive integers a and b that are relative prime, and is concerned with the set of positive numbers NR that have no representation by the linear form ax+by in nonnegative integers x and y. We give a complete characterization of the set NR, and use it to establish a relation between the power sums over its elements and the power sums over the natural numbers. This relation is used to derive new recurrences for the Bernoulli numbers. 相似文献
5.
In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and Bernoulli numbers, including the Miki identity. 相似文献
6.
Takashi Agoh 《Discrete Mathematics》2009,309(4):887-274
Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities. 相似文献
7.
H. Alzer 《Archiv der Mathematik》2000,74(3):207-211
We determine the best possible real constants a\alpha and b\beta such that the inequalities [(2(2n)!)/((2p)2n)] [1/(1-2a-2n)] \leqq |B2n| \leqq [(2(2n)!)/((2p)2n)] [1/(1-2b-2n)]{2(2n)! \over(2\pi)^{2n}} {1 \over 1-2^{\alpha -2n}} \leqq |B_{2n}| \leqq {2(2n)! \over (2\pi )^{2n}}\, {1 \over 1-2^{\beta -2n}}hold for all integers n\geqq 1n\geqq 1. Here, B2, B4, B6,... are Bernoulli numbers. 相似文献
8.
On Miki's identity for Bernoulli numbers 总被引:1,自引:0,他引:1
Ira M. Gessel 《Journal of Number Theory》2005,110(1):75-82
We give a short proof of Miki's identity for Bernoulli numbers,
9.
Mathematical Notes - 相似文献
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11.
Inspired by Borwein et al. (Am. Math. Mon., 116(5):387–412, 2009), we define a sequence of q-analogues for the Bernoulli numbers under the framework of Strodt operators. We show that they not only satisfy identities similar to those of the q-analogue proposed by Carlitz (Duke Math. J., 15(4):987–1000, 1948), but also interesting analytical properties as functions of q. In particular, we give a simple analytic proof of a generalization of an explicit formula for the Bernoulli numbers given by Woon (Math. Mag., 70(1):51–56, 1997). We also define a set of q-analogues for the Stirling numbers of the second kind within our framework and prove a q-extension of a related, well-known closed form relating Bernoulli and Stirling numbers. 相似文献
12.
Analysis Mathematica - Let $$left{ {{d_k}} right}_{k = 1}^infty $$ be an upper-bounded sequence of positive integers and let δE be the uniformly discrete probability measure on the finite... 相似文献
13.
Hao Pan 《Journal of Combinatorial Theory, Series A》2006,113(1):156-175
Using the finite difference calculus and differentiation, we obtain several new identities for Bernoulli and Euler polynomials; some extend Miki's and Matiyasevich's identities, while others generalize a symmetric relation observed by Woodcock and some results due to Sun. 相似文献
14.
Fredholm's integral equation theory and Mittag-Leffler expansion is used for getting characteristic functions and three ways of expressing the Bernoulli numbers. 相似文献
15.
Takashi Agoh 《manuscripta mathematica》1988,61(1):1-10
The main purpose of this paper is to investigate some basic relations (e.g., Voronoi's and Kummer's congruences) of Bernoulli and Euler numbers by manipulating Euler factors in a natural way. 相似文献
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17.
Ayşegül Gökhan Mikail Et Mohammad Mursaleen 《Mathematical and Computer Modelling》2009,49(3-4):548-555
The purpose of this paper is to introduce the concepts of almost lacunary statistical convergence and strongly almost lacunary convergence of sequences of fuzzy numbers. We give some relations related to these concepts. We establish some connections between strongly almost lacunary convergence and almost lacunary statistical convergence of sequences of fuzzy numbers. It is also shown that if a sequence of fuzzy numbers is strongly almost lacunary convergent with respect to an Orlicz function then it is almost lacunary statistical convergent. 相似文献
18.
We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals. Furthermore two new identities on Bernoulli numbers are derived. 相似文献
19.
Ciro D'aniello 《Rendiconti del Circolo Matematico di Palermo》1994,43(3):329-332
In modo elementare si migliorano sensibilmente alcune diseguaglianze recentemente stabilite da D. J. Leeming sui numeri di
Bernoulli [3]. 相似文献
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