Generalized Bernoulli Numbers Research Articles

Class Number Formula for Certain Imaginary Quadratic Fields

In this note we shall show how Carlitz in 1954 could have reached an analogue of the Voronoi congruence in the more difficult case of p≡1(mod4): h(-4p) ≡B(p+1)/2(x4)(mod p), where B(p+1)/2(x4) is the generalized Bernoulli number with x4 being the Kronecker symbol associated to the Gaussian field Q(√-4).

Generalized Bernoulli Polynomials and Numbers, Revisited

We describe with some new details the connection between generalized Bernoulli polynomials, Bernoulli polynomials and generalized Bernoulli numbers (Norlund polynomials). A new recursive and explicit formulae for these polynomials are derived.

Ramanujan type congruences for modular forms of several variables

We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the (k−1)th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight k. We will conclude by giving numerical examples for each case.

On the Gauss sums and generalized Bernoulli numbers

Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas.

On a factorization of Riemann’s ζ function with respect to a quadratic field and its computation

Let K be a quadratic field, and let ζ K its Dedekind zeta function. In this paper we introduce a factorization of ζ K into two functions, L 1 and L 2, defined as partial Euler products of ζ K , which lead to a factorization of Riemann’s ζ function into two functions, p 1 and p 2. We prove that these functions satisfy a functional equation which has a unique solution, and we give series of very fast convergence to them. Moreover, when Δ K > 0 the general term of these series at even positive integers is calculated explicitly in terms of generalized Bernoulli numbers.

On some new congruences for generalized Bernoulli numbers

On some new congruences for generalized Bernoulli numbers

The class number of $\mathbb{Q}(\sqrt{-p})$ and digits of $1/p$

Let p be a prime number such that p ≡ 1 (mod r) for some integer r > 1. Let g > 1 be an integer such that g has order r in (Z/pZ) *. Let be the g-adic expansion. Our result implies that the average digit in the g-adic expansion of 1/p is (g — 1)/2 with error term involving the generalized Bernoulli numbers B 1,χ (where χ is a character modulo p of order r with χ(―1) = ―1). Also, we study, using Bernoulli polynomials and Dirichlet L-functions, the set equidistribution modulo 1 of the elements of the subgroup H n of (ℤ/nℤ) * as n → ∞ whenever |H n |/√n → ∞.

On Witten multiple zeta-functions associated with semisimple Lie algebras II

We prove asymptotic formulas for mean square values of the Euler double zeta-function $\zeta_2(s_0,s)$, with respect to $\Im s$. Those formulas enable us to propose a double analogue of the Lindel{\"o}f hypothesis.

Cuspidal class number of the tower of modular curves X 1(Np n )

We consider a cuspidal class number, which is the order of a subgroup of the full cuspidal divisor class group of X 1(Np n ) with $${p\nmid N}$$ and n ≥ 1. By studying the second generalized Bernoulli numbers, we obtain results similar to ones (Ferrero and Washington in Ann Math (2) 109(2):377–395, 1979; Washington in Invent Math 49:87–97, 1978) about the relative class numbers of cyclotomic $${\mathbb{Z}_p}$$ -extension of an abelian number field.

Generalized higher order Bernoulli number pairs and generalized Stirling number pairs

From a delta series f ( t ) and its compositional inverse g ( t ) , Hsu defined the generalized Stirling number pair ( S ˆ ( n , k ) , s ˆ ( n , k ) ) . In this paper, we further define from f ( t ) and g ( t ) the generalized higher order Bernoulli number pair ( B ˆ n ( z ) , b ˆ n ( z ) ) . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours.

Some Identities of Symmetry for the Generalized Bernoulli Numbers and Polynomials

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order

We give some interesting relationships between the power sums and the generalized Bernoulli numbers attached to of higher order using multivariate -adic invariant integral on .

Sums of products of Cauchy numbers

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.

Bernoulli numbers and zeros of $p$-adic $L$-functions

Rational $p$-adic zeros of the Leopoldt-Kubota $p$-adic $L$-functions give rise to certain sequences of generalized Bernoulli numbers tending $p$-adically to zero, and conversely. This relationship takes different forms depending on whether the corresponding Iwasawa $\lambda$-invariant is one or greater than one. To understand the relationship better it is useful to consider approximate zeros of those functions.

Hybrid mean value on the difference between an integer and its inverse modulo q

The main purpose of this paper is to study the asymptotic property of one class of number-theoretic functions and give a sharp hybrid mean-value formula by using the generalized Bernoulli numbers, Gauss sums and the mean-value theorems of Dirichlet L-functions.

On a [formula omitted]-adic interpolation function for the [formula omitted]-extension of the generalized Bernoulli polynomials and its derivative

In [T. Kim, S.H. Rim, Generalized Carlitz’s q -Bernoulli numbers in the p -adic number field, Adv. Stud. Contemp. Math. 2 (2000) 9–19], the new q -extension of Bernoulli polynomials and generalized Bernoulli numbers attached to χ were constructed by using p -adic invariant integral on Z p . In this paper we construct the new q -extension of generalized Bernoulli polynomials attached to χ due to author and derive the existence of a specific p -adic interpolation function which interpolates the q -extension of generalized Bernoulli polynomials at negative integers. Finally, we give the values of partial derivative for this function and investigate some properties which are related to this interpolation function.

Congruences for Generalized -Bernoulli Polynomials

In this paper, we give some further properties of -adic - -function of two variables, which is recently constructed by Kim (2005) and Cenkci (2006). One of the applications of these properties yields general classes of congruences for generalized -Bernoulli polynomials, which are -extensions of the classes for generalized Bernoulli numbers and polynomials given by Fox (2000), Gunaratne (1995), and Young (1999, 2001).

HYBRID MEAN VALUE OF GENERALIZED BERNOULLI NUMBERS, GENERAL KLOOSTERMAN SUMS AND GAUSS SUMS

The main purpose of this paper is to use the properties of primitive characters, Gauss sums and Ramanujan's sum to study the hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, and give two asymptotic formulae.

Mean Value on the Difference Between a Quadratic Residue and Its Inverse Modulo p

The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions to study the asymptotic property of the difference between a quadratic residue and its inverse modulo p (a prime), and to give an interesting hybrid mean value formula.

An estimate for the powers of the generating function of the Bernoulli numbers

Let t ≥ 0 be real and let e (t) denote the integal , which is the integral analogue to the power series of the exponential function exp(t). Then, by using the Laplace transform Δ(x) of the function δ(t):= exp(t) − e (t) for all real t ≠ 0, we will get the estimate 0 < B n (t) − PP 0[B n ](t) < 1, which is sharp at t = ±∞, where B n is defined by B n (t):= (1 − exp(−t))−n and PP 0[B n ] denotes the principal part of the Laurent expansion of B n at t = 0. We will see that the expression between the two ‘<’ signs strictly increases (on R) and will express its limit at t = 0 in terms of the ‘generalized Bernoulli numbers’ , i.e. by the coefficients of the Taylor expansion of [z/(exp(z) − 1)] n at z = 0.