Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ 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
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ 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.

     

On a relationship for Bernoulli numbers

Mathematical Notes -     

On a q-analogue for Bernoulli numbers

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.     

Characteristic functions and Bernoulli numbers

Fredholm's integral equation theory and Mittag-Leffler expansion is used for getting characteristic functions and three ways of expressing the Bernoulli numbers.     

On Bernoulli and Euler numbers

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.     

Bernoulli numbers and sums of powers

There have been many studies of Bernoulli numbers since Jakob Bernoulli first used the numbers to compute sums of powers, 1 ^p + 2 ^p + 3 ^p + ··· + n^p , where n is any natural number and p is any non-negative integer. By examining patterns of these sums for the first few powers and the relation between their coefficients and Bernoulli numbers, the author hypothesizes and proves a new recursive algorithm for computing Bernoulli numbers, sums of powers, as well as m-ford sums of powers, which enrich the existing literatures of Bernoulli numbers.     

On a formal analogue of the Bernoulli numbers

Using properties of the modular forms G_k, it is shown that G_K(z) = α_kω^2k where z is an integer in an imaginary quadratic field and α_k is algebraic and involves analogues of the Bernoulli numbers. A recursion formula (3) is given for these numbers.     

p-adic zeta functions and Bernoulli numbers

One gives a new proof to the Leopoldt-Kubota-Iwasawa theorem regarding the possibility of the p-adic interpolation of the values of the Riemann zeta-function and of the Dirichlet L-functions at negative integral points. To this end, for each root ? ≠ 1 of unity one introduces and one investigates the numbers C_n(?) which arise in the expansion $$\frac{{\varepsilon - 1}}{{\varepsilon e^z - 1}} = \sum\limits_{n = 0}^\infty {\frac{{C_n (\varepsilon )}}{{n!}}Z^n }$$ One proves a generalization of the Kummer congruences for the Bernoulli numbers.     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

On hypergeometric Bernoulli numbers and polynomials

We provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.     

The Coxeter polynomial for a one point extension algebra

Let Λ be a finite-dimensional algebra over an algebraically closed field k of finite global dimension. Let M be a finitely generated Λ-module and let $\Gamma =\Lambda [M]$ be the one point extension algebra. We show how to compute the Coxeter polynomial for Γ from the Coxeter polynomial of Λ and homological invariants of M.     

Arithmetical properties of hypergeometric Bernoulli numbers

In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers.