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Poly-Bernoulli numbers

1997
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Journal de Théorie des Nombres de Bordeaux
*

By using polylogarithm series, we define "poly-

doi:10.5802/jtnb.197
fatcat:gm7xxt2xx5gjddpwjbmb2jpdsu
*Bernoulli**numbers*" which generalize classical*Bernoulli**numbers*. ... a sort of duality for negative index poly-*Bernoulli**numbers*. ...##
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Poly-Bernoulli Numbers
[chapter]

2014
*
Springer Monographs in Mathematics
*

By using polylogarithm series, we define "poly-

doi:10.1007/978-4-431-54919-2_14
fatcat:lae45i7vhnbjlgepmpmukbwsha
*Bernoulli**numbers*" which generalize classical*Bernoulli**numbers*. ... a sort of duality for negative index poly-*Bernoulli**numbers*. ...##
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Compositional Bernoulli numbers
[article]

2008
*
arXiv
*
pre-print

We define and study the combinatorial properties of compositional

arXiv:0708.0809v2
fatcat:al6ynx7afrfxbnny3rdivbsndm
*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! ...##
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Congruences concerning Bernoulli numbers and Bernoulli polynomials

2000
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Discrete Applied Mathematics
*

We also establish similar congruences for generalized

doi:10.1016/s0166-218x(00)00184-0
fatcat:uxygqhrdqzgejnxinziyl3piee
*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). ...##
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Poly-Bernoulli Numbers and Eulerian Numbers
[article]

2018
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arXiv
*
pre-print

In this note we prove combinatorially some new formulas connecting poly-

arXiv:1804.01868v1
fatcat:d3nmox4cvffg7pl2odhhnjzahi
*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 . ...##
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Numbers Related to Bernoulli-Goss Numbers

2014
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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

doi:10.12691/tjant-2-1-4
fatcat:hggyecrrvfg4tkolpgocffrkku
*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 ...##
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The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked

2019
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Applied Mathematics
*

To complete the work we propose tables for calculating in easiest manners possibly the

doi:10.4236/am.2019.103009
fatcat:rzoevspuz5exvbdnluwxm76n4i
*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 ...##
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q-Bernoulli numbers and q-Bernoulli polynomials revisited

2011
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Advances in Difference Equations
*

This paper performs a further investigation on the q-

doi:10.1186/1687-1847-2011-33
fatcat:ckmlsivszzfi3hzjh3libdjosq
*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. ...##
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On Bernoulli numbers, II

1982
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Journal of Number Theory
*

We remark that Ramanujan's conjecture that the numerator of Bn/2n in its lowest terms is a prime

doi:10.1016/0022-314x(82)90057-9
fatcat:2wcmqbmlbbbtbawmsj4xkcx3vq
*number*, fails already for n = 10. In fact B,, 174611 283. 617 -= 20 20.330=20+3.5.11' ...##
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Bernoulli Numbers and Solitons

2005
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Journal of Nonlinear Mathematical Physics
*

We present a new formula for the

doi:10.2991/jnmp.2005.12.4.3
fatcat:ujve6hoikjd65eq526qe4tnieq
*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 ...##
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More about Bernoulli Numbers

2016
*
Journal of Mathematics and System Science
*

And based on the redundancy, a process for obtaining the

doi:10.17265/2159-5291/2016.03.005
fatcat:zcc5rdyiofdjvdwofmsiwjccpy
*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. ...##
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Bernoulli Numbers & Vandermonde Matrices

2016
*
Journal of Mathematics and System Science
*

Abstracts: Matrix structuring is a very beautiful way to place

doi:10.17265/2159-5291/2016.02.006
fatcat:3w7rgor4czaljdled64pql5neu
*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. ...##
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q-Bernoulli numbers and Stirling numbers(2)
[article]

2007
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arXiv
*
pre-print

In this paper we study q-

arXiv:0710.4976v1
fatcat:fe4rq63lpremti5gdgha7j4qea
*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*. ...##
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Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index
[article]

2015
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arXiv
*
pre-print

We researched relations between Multi-Poly-

arXiv:1503.04933v1
fatcat:gn3thwl25vbxblscmenl6hbhju
*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*...##
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On the generalized Bernoulli numbers
[article]

2018
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arXiv
*
pre-print

We derive an expression for the generalized

arXiv:1801.08027v1
fatcat:b7dlgmluh5gs3pk4ideaqnijma
*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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