Congruences concerning Bernoulli numbers and Bernoulli polynomials

Sun, Zhi-Hong

doi:10.1016/s0166-218x(00)00184-0

Cited by 125 publications

(104 citation statements)

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“…By Theorem 3.12, we have (23) and Theorem 4.14 (120). By Theorem 4.6, we have (23) and Theorem 4.14 (119). Therefore, we have the equality (145) by the equality (146).…”

Section: Definitions and Functional Equations Of Finite Multiple Polymentioning

confidence: 91%

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On functional equations of finite multiple polylogarithms

Sakugawa

Seki

2017

Journal of Algebra

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Abstract. Recently, several people study finite multiple zeta values (FMZVs) and finite polylogarithms (FPs). In this paper, we introduce finite multiple polylogarithms (FMPs), which are natural generalizations of FMZVs and FPs, and we establish functional equations of FMPs. As applications of these functional equations, we calculate special values of FMPs containing generalizations of congruences obtained by Meštrović, Z. W. Sun, L. L. Zhao, Tauraso, and J. Zhao. We show supercongruences for certain generalized Bernoulli numbers and the Bernoulli numbers as an appendix. IntroductionFrom the end of twentieth century to the beginning of twenty-first century, Hoffman and J. Zhao had started research about mod p multiple harmonic sums, which are motivated by various generalizations of classical Wolstenholme's theorem. Recently, Kaneko and Zagier introduced a new "adélic" framework to describe the pioneer works by Hoffman and Zhao and they defined finite multiple zeta values (FMZVs). Let k 1 , . . . , k m be positive integers and k := (k 1 , . . . , k m ). Definition 1.1 (Kaneko and Zagier [10,11]). The finite multiple zeta value ζ A (k) is defined byand the finite multiple zeta-star value ζ ⋆ A (k) is defined byHere, the Q-algebra A is defined bywhere p runs over all prime numbers.In this framework, Kaneko and Zagier established a conjecture, which states that there is an isomorphism between the Q-algebra spanned by FMZVs and the quotient Q-algebra modulo the ideal generated by ζ(2) of the Q-algebra spanned by the usual multiple zeta values.On

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“…By Theorem 3.12, we have (23) and Theorem 4.14 (120). By Theorem 4.6, we have (23) and Theorem 4.14 (119). Therefore, we have the equality (145) by the equality (146).…”

Section: Definitions and Functional Equations Of Finite Multiple Polymentioning

confidence: 91%

“…The equalities (23), (26), (27), (28), and (29) Now, we define the finite multiple polylogarithms which are main objects in this paper.…”

Section: Definitions and Functional Equations Of Finite Multiple Polymentioning

confidence: 99%

On functional equations of finite multiple polylogarithms

Sakugawa

Seki

2017

Journal of Algebra

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“…Let p > 3 be a prime, n ≤ p − 2 a positive integer, Lemma 2 (cf. [2], [3], [4] or [5]). Let p be an odd prime> 3 and let r be a positive integer, r ≤ p − 3.…”

Section: Introductionmentioning

confidence: 99%

A generalization of a curious congruence on harmonic sums

Zhou

Cai

2006

Proc. Amer. Math. Soc.

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“…In this proof we additionally use certain congruences by H. Pan (Lemma 2.4) which have been derived in [26] via combinatorial methods. We also use some congruences (Lemma 2.5) which were proved by Z. H. Sun in [34] via a standard technique expressing sum of powers in terms of Bernoulli numbers.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Congruences involving the Fermat quotient

Meštrović

2013

Czech Math J

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Congruences concerning Bernoulli numbers and Bernoulli polynomials

Cited by 125 publications

References 10 publications

On functional equations of finite multiple polylogarithms

On functional equations of finite multiple polylogarithms

A generalization of a curious congruence on harmonic sums

Congruences involving the Fermat quotient

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