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Congruences for Generalized -Bernoulli Polynomials

Mehmet Cenkci, Veli Kurt
2008 Journal of Inequalities and Applications  
One of the applications of these properties yields 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.  ... 
doi:10.1155/2008/270713 fatcat:rz2gyvat7ffkzpvxwu7wnm452y

General congruences for Bernoulli polynomials

Zhi-Wei Sun
2003 Discrete Mathematics  
In this paper we establish some explicit 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.  ... 
doi:10.1016/s0012-365x(02)00504-6 fatcat:r4siteqbqbaavjkyro7ge4oag4

Kummer congruences for expressions involving generalized Bernoulli polynomials

Glenn J. Fox
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, 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  ... 
doi:10.5802/jtnb.353 fatcat:l5vdeyr7djgpralvwxuwf7i334

The Lazard formal group, universal congruences and special values of zeta functions

Piergiulio Tempesta
2015 Transactions of the American Mathematical Society  
In particular, it is shown how to construct 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.  ... 
doi:10.1090/tran/6234 fatcat:4ckghnvijveyfaiyqcyzpe3hya

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

Anthony G. Shannon, Ömür Deveci, Özgűr Erdağ
2019 Notes on Number Theory and Discrete Mathematics  
Relationships, in terms of equations and 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  ... 
doi:10.7546/nntdm.2019.25.1.193-198 fatcat:gikwichgl5clvbjtptezinelva

L-series and Hurwitz zeta functions associated with the universal formal group

Tempesta Pierluigi
2010 Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V  
The properties of the universal 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.  ... 
doi:10.2422/2036-2145.2010.1.05 fatcat:ifztuwakorf2jpfhvdpoi7exye

Bernoulli Numbers and Generalized Factorial Sums

Paul Thomas Young
2011 Integers  
These are derived from an expression 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.  ... 
doi:10.1515/integ.2011.041 fatcat:cpw4ktpwezdj3d2zc773hrfcuy

On multiple zeros of Bernoulli polynomials

Karl Dilcher
2008 Acta Arithmetica  
Closely related to the 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.  ... 
doi:10.4064/aa134-2-6 fatcat:wvwlaj52pza4djrc5hlgukpzlm

A general family of congruences for Bernoulli numbers

Julian Rosen
2018 International Journal of Number Theory  
We prove a 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.  ... 
doi:10.1142/s1793042118501129 fatcat:kab3kinmsfacnjffdrts2ke5mi

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).  ... 

Congruences ofp-adic Integer Order Bernoulli Numbers

Arnold Adelberg
1996 Journal of Number Theory  
In this paper we establish some new 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 .  ... 
doi:10.1006/jnth.1996.0103 fatcat:ldjyn5zxyvdqbaikykf7bmc3l4

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.  ... 

Polylogarithms, hyperfunctions and generalized Lipschitz summation formulae [article]

Stefano Marmi, Piergiulio Tempesta
2007 arXiv   pre-print
It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of 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.  ... 
arXiv:0712.1046v1 fatcat:zw26nfa2gbccpcyjifc3o5ld7q
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