Geodesic
Some Congruences for Generalized Euler Numbers
Canadian journal of mathematics, Tome 35 (1983) no. 4, pp. 687-709

Voir la notice de l'article provenant de la source Cambridge University Press

The generalized Euler numbers may be defined by Since is zero unless m divides n, we shall write for . Leeming and MacLeod [12] recently gave some congruences for these numbers. They found congruences (mod 16) for where m = 3, 6, 8, 12, and 16. Thus for m = 3, their congruence is They also proved that , and , and they made several conjectures which may be stated as follows: C1 C2 C3 C4 
Gessel, Ira M. Some Congruences for Generalized Euler Numbers. Canadian journal of mathematics, Tome 35 (1983) no. 4, pp. 687-709. doi: 10.4153/CJM-1983-039-5
@article{10_4153_CJM_1983_039_5,
     author = {Gessel, Ira M.},
     title = {Some {Congruences} for {Generalized} {Euler} {Numbers}},
     journal = {Canadian journal of mathematics},
     pages = {687--709},
     year = {1983},
     volume = {35},
     number = {4},
     doi = {10.4153/CJM-1983-039-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-039-5/}
}
TY  - JOUR
AU  - Gessel, Ira M.
TI  - Some Congruences for Generalized Euler Numbers
JO  - Canadian journal of mathematics
PY  - 1983
SP  - 687
EP  - 709
VL  - 35
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-039-5/
DO  - 10.4153/CJM-1983-039-5
ID  - 10_4153_CJM_1983_039_5
ER  - 
%0 Journal Article
%A Gessel, Ira M.
%T Some Congruences for Generalized Euler Numbers
%J Canadian journal of mathematics
%D 1983
%P 687-709
%V 35
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1983-039-5/
%R 10.4153/CJM-1983-039-5
%F 10_4153_CJM_1983_039_5

[1] 1. Brun, V., Stubban, J. O., Fjeldstad, J. E., Lyche, R. Tambs, Aubert, K. E., Ljunggren, W., and Jacobsthal, E., On the divisibility of the difference between two binomial coefficients. Den l lte Skandinaviske Matematikerkongress, Trondheim (1949), 42–54. Google Scholar

[2] 2. Carlitz, L., Some arithmetic properties of the Olivier functions, Math. Annalen 128 (1954-1955), 412–419. Google Scholar

[3] 3. Carlitz, L., A note on Rummer's congruences, Arch. Math. 7 (1957), 441–445. Google Scholar

[4] 4. Carlitz, L., Kummer's congruences (mod 2r), Monatshefte fur Math. 63 (1959), 394–400. Google Scholar

[5] 5. Carlitz, L., Composition of sequences satisfying Kummefs congruences, Collect. Math. 11 (1959), 137–152. Google Scholar

[6] 6. Carlitz, L., Some arithmetic properties of a special sequence of integers, Can. Math. Bull. 19 (1976), 425–429. Google Scholar

[7] 7. Frobenius, F. G., Uber die Bernoullischen Zahlen und die Eulerschen Polynôme, Gesammelte Abhandlungen III (Springer, Berlin-Heidelberg-New York, 1968), 440–478. Originally published in Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1910), 809–847. Google Scholar

[8] 8. Gessel, I. M., Some congruences for Apery numbers, J. Number Theory 14 (1982), 362–368. Google Scholar

[9] 9. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 4th ed. (Oxford University Press, 1965). Google Scholar

[10] 10. Kazandzidis, G. S., On congruences in number-theory, Bull. Soc. Math. Grèce (N. S.) 10 (1969), 35–40. Google Scholar

[11] 11. Kummer, E. F., Uber eine allgemeine Eigenschaft der rationalen Entwickelungscoëfficienten einer bestimmten Gattung analytischer Function, J. reine angew. Math. 41 (1851), 368–372. Google Scholar

[12] 12. Leeming, D. J. and MacLeod, R. A., Some properties of generalized Euler numbers, Can. J. Math. 33 (1981), 606–617. Google Scholar

[13] 13. Rota, G.-C. and Sagan, B., Congruences derived from group action, Europ. J. Combinatorics 1 (1980), 67–76. Google Scholar

[14] 14. Smith, J. H., Combinatorial congruences derived from the action of Sylow subgroups of the symmetric group, preprint. Google Scholar

[15] 15. Stevens, H., Generalized Kummer congruences for products of sequences, Duke Math. J. 28 (1961), 25–38. Google Scholar

[16] 16. Stevens, H., Generalized Kummer congruences for the products of sequences. Applications, Duke Math. J. 28 (1961), 261–275. Google Scholar

[17] 17. Trakhtman, Yu. A., On the divisibility of certain differences formed from binomial coefficients (Russian), Doklady Akad. Nauk Arm. S. S. R. 59 (1974), 10–16. Google Scholar

Cité par Sources :