Kummer congruences for expressions involving generalized Bernoulli polynomials

Glenn J. Fox

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 1, page 187-204
  • ISSN: 1246-7405

Abstract

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We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.

How to cite

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Fox, Glenn J.. "Kummer congruences for expressions involving generalized Bernoulli polynomials." Journal de théorie des nombres de Bordeaux 14.1 (2002): 187-204. <http://eudml.org/doc/248903>.

@article{Fox2002,
abstract = {We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.},
author = {Fox, Glenn J.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Bernoulli numbers; -adic -functions; congruences; generalized Bernoulli polynomials},
language = {eng},
number = {1},
pages = {187-204},
publisher = {Université Bordeaux I},
title = {Kummer congruences for expressions involving generalized Bernoulli polynomials},
url = {http://eudml.org/doc/248903},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Fox, Glenn J.
TI - Kummer congruences for expressions involving generalized Bernoulli polynomials
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 187
EP - 204
AB - We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.
LA - eng
KW - Bernoulli numbers; -adic -functions; congruences; generalized Bernoulli polynomials
UR - http://eudml.org/doc/248903
ER -

References

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  2. [2] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, Revised and enlarged edition. D. Reidel Publishing Co., Dordrecht, 1974. Zbl0283.05001MR460128
  3. [3] M. Eie, Y.L. Ong, A generalization of Kummer's congruences. Abh. Math. Sem. Univ. Hamburg67 (1997), 149-157. Zbl0896.11035MR1481532
  4. [4] G.J. Fox, A p-adic L-function of two variables. Enseign. Math. (2) 46 (2000), 225-278. Zbl0999.11073MR1805401
  5. [5] J. Fresnel, Nombres de Bernoulli et fonctions L p-adiques. Ann. Inst. Fourier17 (1967), 281-333. Zbl0157.10302MR224570
  6. [6] H.S. Gunaratne, A new generalisation of the Kummer congruence. Computational algebra and number theory (Sydney, 1992), Math. Appl.325, Kluwer Acad. Publ., Dordrecht, 1995, 255-265. Zbl0833.11005MR1344935
  7. [7] H.S. Gunaratne, Periodicity of Kummer congruences, Number theory. (Halifax, NS,1994), CMS Conf. Proc. 15, Amer. Math. Soc., Providence, RI, 1995, 209-214. Zbl0843.11012MR1353933
  8. [8] T. Kubota, H.-W. Leopoldt, Eine p-adische Theorie der Zetawerte. 1. Einführung der p-adischen Dirichdetschen L-funktionen. J. Reine Angew. Math.214/215 (1964), 328-339. Zbl0186.09103MR163900
  9. [9] W. Narkiewicz, Polynomial mappings. Lecture Notes in Mathematics, 1600, Springer-Verlag, Berlin, 1995. Zbl0829.11002MR1367962
  10. [10] K. Shiratani, Kummer's congruence for generalized Bernoulli numbers and its application. Mem. Fac. Sci. Kyushu Univ. Ser. A26 (1972), 119-138. Zbl0243.12009MR360528
  11. [11] L.C. Washington, Introduction to Cyclotomic Fields, Second edition. Graduate Texts in Mathematics83, Springer-Verlag, New York, 1997. Zbl0966.11047MR1421575
  12. [12] P.T. Young, Congruences for Bernoulli, Euler, and Stirling numbers. J. Number Theory78 (1999), 204-227. Zbl0939.11014MR1713481
  13. [13] P.T. Young, Kummer congruences for values of Bernouili and Euler polynomials. Acta Arith.99 (2001), 277-288. Zbl0982.11008MR1845351

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