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Congruences for Generalized -Bernoulli Polynomials

2008
*
Journal of Inequalities and Applications
*

One of the applications of these properties yields

doi:10.1155/2008/270713
fatcat:rz2gyvat7ffkzpvxwu7wnm452y
*general*classes of*congruences**for**generalized*q-*Bernoulli**polynomials*, which are qextensions of the classes*for**generalized**Bernoulli*numbers and*polynomials*...*for**generalized*q-*Bernoulli**polynomials*. ... Fox 6 derived*congruences*similar to those above*for*the*generalized**Bernoulli**polynomials*without restrictions on the character χ. ...##
###
General congruences for Bernoulli polynomials

2003
*
Discrete Mathematics
*

In this paper we establish some explicit

doi:10.1016/s0012-365x(02)00504-6
fatcat:r4siteqbqbaavjkyro7ge4oag4
*congruences**for**Bernoulli**polynomials*modulo a*general*positive integer. In particular Voronoi's and Kummer's*congruences*are vastly extended. ... Acknowledgements The author is indebted to the referees*for*their many helpful suggestions. ...*Bernoulli**polynomials*have many applications, they are of independent interest as well. In this paper we aim to give explicit*congruences**for**Bernoulli**polynomials*modulo a*general*positive integer. ...##
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Kummer congruences for expressions involving generalized Bernoulli polynomials

2002
*
Journal de Théorie des Nombres de Bordeaux
*

Acknowledgements Finally, the author would like to express his gratitude to his graduate advisor, Professor Andrew Granville of the University of Georgia, Athens,

doi:10.5802/jtnb.353
fatcat:l5vdeyr7djgpralvwxuwf7i334
*for*suggesting the consideration of this ... ~(0), and, thus, provides equivalences between*congruences*on /3n,x(r), involving*generalized**Bernoulli**polynomials*, and*congruences*on /3n,x(O), involving*generalized**Bernoulli*numbers. ... The*generating*function*for*the classical*Bernoulli**polynomials*, is given by and from the classical*Bernoulli**polynomials*we obtain the classical*Bernoulli*numbers, Bn = The*Bernoulli*numbers, Bn, are ...##
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The Lazard formal group, universal congruences and special values of zeta functions

2015
*
Transactions of the American Mathematical Society
*

In particular, it is shown how to construct

doi:10.1090/tran/6234
fatcat:4ckghnvijveyfaiyqcyzpe3hya
*general*Almkvist--Meurman--type*congruences**for*the universal*Bernoulli**polynomials*that are related with the Lazard universal formal group Tempesta1-Tempesta3 ... New*congruences*are also obtained, useful to compute special values of a new class of Riemann--Hurwitz--type zeta functions. ... In this paper, we address the following*general*problem: Construct*congruences**for*the Universal*Bernoulli**polynomials*and apply them to compute special values of the related zeta functions. ...##
###
Page 8971 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

(PRC-NAN; Nanjing)

*General**congruences**for**Bernoulli**polynomials*. (English summary) Discrete Math. 262 (2003), no. 1-3, 253-276. ... The author obtains very*general*explicit*congruences**for**Bernoulli*2003m: 1 1040 numbers and other values of*Bernoulli**polynomials*modulo gen- eral positive integers. ...##
###
Generalized Fibonacci numbers and Bernoulli polynomials

2019
*
Notes on Number Theory and Discrete Mathematics
*

Relationships, in terms of equations and

doi:10.7546/nntdm.2019.25.1.193-198
fatcat:gikwichgl5clvbjtptezinelva
*congruences*, are developed between the*Bernoulli*numbers and arbitrary order*generalizations*of the ordinary Fibonacci and Lucas numbers. ...*Bernoulli**polynomial**congruences*[7] , so there are too*for*the Fibonacci*polynomials*. ... Introduction A*generating*function*for*Fibonacci*polynomials*{ } ) (x u n [3, 4] was demonstrated in [3] where the j P are arbitrary integers and the j α are the distinct roots of the auxiliary equation ...##
###
L-series and Hurwitz zeta functions associated with the universal formal group

2010
*
Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V
*

The properties of the universal

doi:10.2422/2036-2145.2010.1.05
fatcat:ifztuwakorf2jpfhvdpoi7exye
*Bernoulli**polynomials*are illustrated and a new class of related L-functions is constructed. ... A*generalization*of the Riemann-Hurwitz zeta function is also proposed. (2010) : 11M41 (primary); 55N22 (secondary). ... More*general*versions of these*congruences**for*the classical*Bernoulli*numbers are known in the literature [28] . As shown by Adelberg, the numbers B n satisfy a universal*congruence*. ...##
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Bernoulli Numbers and Generalized Factorial Sums

2011
*
Integers
*

These are derived from an expression

doi:10.1515/integ.2011.041
fatcat:cpw4ktpwezdj3d2zc773hrfcuy
*for*the Mahler coefficients of degenerate*Bernoulli*numbers. As corollaries several unusual identities and*congruences*are derived. ... AbstractWe prove a pair of identities expressing*Bernoulli*numbers and*Bernoulli*numbers of the second kind as sums of*generalized*falling factorials. ... We may also derive the following*congruences**for*the*Bernoulli*numbers. These are*generalizations*of the well-known fact that pB p−1 ≡ −1 (mod p)*for*odd primes p. Corollary 2. ...##
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On multiple zeros of Bernoulli polynomials

2008
*
Acta Arithmetica
*

Closely related to the

doi:10.4064/aa134-2-6
fatcat:wvwlaj52pza4djrc5hlgukpzlm
*Bernoulli**polynomials*are the Euler*polynomials*E n (x) which can be defined by the*generating*function 2e xt e t + 1 = ∞ n=0 E n (x) t n n! ... We begin by quoting some basic properties of*Bernoulli**polynomials*,*for*easy reference. ...##
###
A general family of congruences for Bernoulli numbers

2018
*
International Journal of Number Theory
*

We prove a

doi:10.1142/s1793042118501129
fatcat:kab3kinmsfacnjffdrts2ke5mi
*general*family of*congruences**for**Bernoulli*numbers whose index is a*polynomial*function of a prime, modulo a power of that prime. ... Our family*generalizes*many known results, including the von Staudt--Clausen theorem and Kummer's*congruence*. ... The*congruences*(5) and (6) also follow from Theorem 1.1, as do many of the*congruences**for**Bernoulli*numbers given in [1] . The author does not know whether the converse of Theorem 1.1 holds. ...##
###
Page 2795 of Mathematical Reviews Vol. , Issue 97E
[page]

1997
*
Mathematical Reviews
*

The author presents some

*congruences**for*the p-adic integer- order*Bernoulli*numbers and as a consequence he gets the classical Kummer*congruence**for*the ordinary*Bernoulli*numbers (under a stronger unimportant ... A direct formula*for*evaluating this*general*term is obtained. ...##
###
Page 3790 of Mathematical Reviews Vol. , Issue 80J
[page]

1980
*
Mathematical Reviews
*

They first

*generalize*Voronoi’s*congruence*to a*congruence*modulo p? and then use this to prove a*generalization*of E. Lehmer’s*congruence*modulo p*. ... In particular,*for*n= 1, BO(x)= Bx), the classical*Bernoulli**polynomials*, and set P,(x)= B(x- [x]) ((x] denotes the greatest integer function). ...##
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Congruences ofp-adic Integer Order Bernoulli Numbers

1996
*
Journal of Number Theory
*

In this paper we establish some new

doi:10.1006/jnth.1996.0103
fatcat:ldjyn5zxyvdqbaikykf7bmc3l4
*congruences*of p-adic integer order*Bernoulli*numbers. These*generalize*the Kummer*congruences**for*ordinary*Bernoulli*numbers. ... We apply our*congruences*to prove irreducibility of certain*Bernoulli**polynomials*with order divisible by p and to get new*congruences**for*Stirling numbers. 1996 Academic Press, Inc. ... We also get in Section 5 a new*congruence*mod pl 2*for*B (l ) p when p | l, which*generalizes*a Carlitz*congruence**for*B ( p) p . ...##
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Page 548 of Annals of Mathematics Vol. 39, Issue 3
[page]

1938
*
Annals of Mathematics
*

The

*general**congruences*in §3 are sufficient*for*a detailed investigation of the residue mod p of &%”. ... These*Bernoulli*sums*for*m = 1, --- , 10 are available in many places, and will be used in solving difference*congruences*without further reference. ...##
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Polylogarithms, hyperfunctions and generalized Lipschitz summation formulae
[article]

2007
*
arXiv
*
pre-print

It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of

arXiv:0712.1046v1
fatcat:zw26nfa2gbccpcyjifc3o5ld7q
*Bernoulli*--type*polynomials*. ... A*generalization*of the classical Lipschitz summation formula is proposed. ... The reasons to consider such a*generalization*of*Bernoulli**polynomials*are manifold. ...
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