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### Poly-Bernoulli numbers

Masanobu Kaneko
1997 Journal de Théorie des Nombres de Bordeaux
By using polylogarithm series, we define "poly-Bernoulli numbers" which generalize classical Bernoulli numbers.  ...  a sort of duality for negative index poly-Bernoulli numbers.  ...

### Poly-Bernoulli Numbers [chapter]

Tomoyoshi Ibukiyama, Masanobu Kaneko
2014 Springer Monographs in Mathematics
By using polylogarithm series, we define "poly-Bernoulli numbers" which generalize classical Bernoulli numbers.  ...  a sort of duality for negative index poly-Bernoulli numbers.  ...

### Compositional Bernoulli numbers [article]

Hector Blandin, Rafael Diaz
2008 arXiv   pre-print
We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics.  ...  Combinatorics of Bernoulli numbers We introduce a generalization of Bernoulli numbers which may be motivated as follows.  ...  Similarly one can consider Bernoulli numbers B cos N,n associated with the cosine function for N = 2L an even number. For N = 2 the Bernoulli numbers B cos 2,n are such that: x 2 /2!  ...

### Congruences concerning Bernoulli numbers and Bernoulli polynomials

Zhi-Hong Sun
2000 Discrete Applied Mathematics
We also establish similar congruences for generalized Bernoulli numbers {Bn; }. ?  ...  Let {Bn(x)} denote Bernoulli polynomials.  ...  Introduction The Bernoulli numbers {B n } and Bernoulli polynomials {B n (x)} are deÿned as follows: Let p be an odd prime, and b an even number with b ≡ 0 (mod p − 1).  ...

### Poly-Bernoulli Numbers and Eulerian Numbers [article]

Beata Benyi, Peter Hajnal
2018 arXiv   pre-print
In this note we prove combinatorially some new formulas connecting poly-Bernoulli numbers with negative indices to Eulerian numbers.  ...  As the name indicates, poly-Bernoulli numbers are generalizations of the Bernoulli numbers. For k = 1 B (1) n are the classical Bernoulli numbers with B 1 = 1 2 .  ...  In this note we are concerned only with poly-Bernoulli numbers with negative indices. For convenience, we denote by B n,k the poly-Bernoulli numbers B (−k) n .  ...

### Numbers Related to Bernoulli-Goss Numbers

Mohamed Ould Douh Benough
2014 Turkish Journal of Analysis and Number Theory
In this paper, we generalize a Goss result appeared in ([5], page 325, line 19, for i=1 ), and give a characterization of some numbers of Bernoulli-Goss [5] by introducing the special numbers M(d).  ...  Let ( ) [ ] a T ,a monic n q B n a ∈ = ∑  denotes the n-th Bernoulli-Goss number [5] which is a special value of the zeta function of Goss and is in [ ] q T  .  ...  obtain: ( ) ( ) ( ) 1 1 1 1 1 m m j m j m j m j j m j S m L S m L + + + + + − − − + = − ⇒ − + = Therefore : ( ) ( 1) ij j i j S i L − − = This terminates the proof.  Definition We define the i-th Bernoulli-Goss  ...

### The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked

Do Tan Si
2019 Applied Mathematics
To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.  ...  It follows that ( ) d d d d n z − applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order.  ...  Curiously by replacing z with n and consequently Z with Obtaining Practically Bernoulli Numbers, Bernoulli Functions and Powers Sums Calculations of Bernoulli Numbers For calculating m B we remark  ...

### q-Bernoulli numbers and q-Bernoulli polynomials revisited

Cheon Ryoo, Taekyun Kim, Byungje Lee
This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al.  ...  It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable.  ...  q-Bernoulli numbers and q-Bernoulli polynomials revisited In this section, we perform a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al.  ...

### On Bernoulli numbers, II

P. Chowla, S. Chowla
1982 Journal of Number Theory
We remark that Ramanujan's conjecture that the numerator of Bn/2n in its lowest terms is a prime number, fails already for n = 10. In fact B,, 174611 283. 617 -= 20 20.330=20+3.5.11'  ...

### Bernoulli Numbers and Solitons

Marie-Pierre Grosset, Alexander P Veselov
2005 Journal of Nonlinear Mathematical Physics
We present a new formula for the Bernoulli numbers as the following integral $$B_{2m} =\frac{(-1)^{m-1}}{2^{2m+1}} \int_{-\infty}^{+\infty} (\frac{d^{m-1}}{dx^{m-1}} {sech}^2 x)^2dx.$$ This formula is  ...  motivated by the results of Fairlie and Veselov, who discovered the relation of Bernoulli polynomials with soliton theory.  ...  All odd Bernoulli numbers except B 1 = − 1 2 are zero and the first even Bernoulli numbers are They play an important role in analysis, number theory, algebraic topology and many other areas of mathematics  ...

Nick Huo Han Huang
2016 Journal of Mathematics and System Science
And based on the redundancy, a process for obtaining the Bernoulli Numbers is elaborated.  ...  Abstracts: Bernoulli Numbers are coded with Deterministic Redundancy of Arithmetic Operations, adding and multiplying or exponent, in Natural Number System.  ...  Actually, the Bernoulli Numbers a ij can be obtained by the below iteration process. p=0, b 00 =1, then a 11 =1, i.e. ∑1=n p=1, b 10 =-1, b 11 =2, then a 22 =1/2, a 21 =1/2, i.e.  ...

### Bernoulli Numbers & Vandermonde Matrices

Nick Huo Han Huang
2016 Journal of Mathematics and System Science
Abstracts: Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened.  ...  Natural Numbers are mathematics seeds and Natural Number System (NNS) breeds the whole world mathematically.  ...  p, by iteration process, a complete matrix of Bernoulli numbers can be generated with the below equation.  ...

### q-Bernoulli numbers and Stirling numbers(2) [article]

Taekyun Kim
2007 arXiv   pre-print
In this paper we study q-Bernoulli numbers and polynomials related to q-Stirling numbers.  ...  From thsese studying we investigate some interesting q-stirling numbers' identities related to q-Bernoulli numbers.  ...  [k + 1] q (−1) k s 2 (k, n − k, q), where β m,q are m-th Carlitz q-Bernoulli numbers.  ...

### Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index [article]

Hiroyuki Komaki
2015 arXiv   pre-print
We researched relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index in particular.  ...  In section 3, as main results, we introduce some relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index in particular.  ...  1 1 =1, we obtain B (−1, Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index In this section, we introduce relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli  ...

### On the generalized Bernoulli numbers [article]

Donal F. Connon
2018 arXiv   pre-print
We derive an expression for the generalized Bernoulli numbers in terms of the Bernoulli numbers involving the (exponential) complete Bell polynomials.  ...  Bernoulli numbers n B are defined by the generating function [ the sum is taken over all partitions () r  of r , i.e. over all sets of integers j k such that 1 2 3 23 r k k k rk  ...
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