Kummer congruences for expressions involving generalized Bernoulli polynomials
Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 187-204.

Nous illustrons le fait qu'une expression particulière, impliquant les polynômes de Bernoulli généralisés, satisfait un système de congruences si et seulement si une expression semblable, impliquant les nombres de Bernoulli généralisés, satisfait les mêmes relations de congruence. Parmi ces relations se trouvent les congruences de Kummer ainsi que des généralisations fournies par Gunaratne.

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.

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     title = {Kummer congruences for expressions involving generalized {Bernoulli} polynomials},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {187--204},
     publisher = {Universit\'e Bordeaux I},
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     number = {1},
     year = {2002},
     zbl = {1022.11008},
     mrnumber = {1925997},
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     url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_187_0/}
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Glenn J. Fox. Kummer congruences for expressions involving generalized Bernoulli polynomials. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 187-204. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_1_187_0/

[1] L. Carlitz, Arithmetic properties of generalized Bernoulli numbers. J. Reine Angew. Math. 202 (1959), 174-182. | MR | Zbl

[2] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, Revised and enlarged edition. D. Reidel Publishing Co., Dordrecht, 1974. | MR | Zbl

[3] M. Eie, Y.L. Ong, A generalization of Kummer's congruences. Abh. Math. Sem. Univ. Hamburg 67 (1997), 149-157. | MR | Zbl

[4] G.J. Fox, A p-adic L-function of two variables. Enseign. Math. (2) 46 (2000), 225-278. | MR | Zbl

[5] J. Fresnel, Nombres de Bernoulli et fonctions L p-adiques. Ann. Inst. Fourier 17 (1967), 281-333. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[9] W. Narkiewicz, Polynomial mappings. Lecture Notes in Mathematics, 1600, Springer-Verlag, Berlin, 1995. | MR | Zbl

[10] K. Shiratani, Kummer's congruence for generalized Bernoulli numbers and its application. Mem. Fac. Sci. Kyushu Univ. Ser. A 26 (1972), 119-138. | MR | Zbl

[11] L.C. Washington, Introduction to Cyclotomic Fields, Second edition. Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1997. | MR | Zbl

[12] P.T. Young, Congruences for Bernoulli, Euler, and Stirling numbers. J. Number Theory 78 (1999), 204-227. | MR | Zbl

[13] P.T. Young, Kummer congruences for values of Bernouili and Euler polynomials. Acta Arith. 99 (2001), 277-288. | MR | Zbl