Hostname: page-component-7b7567c8c-lpg2g Total loading time: 0 Render date: 2026-11-15T11:03:32.652Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Congruences for Generalized Euler Numbers

Published online by Cambridge University Press:  20 November 2018

Ira M. Gessel
Ira M. Gessel*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 4 , 01 August 1983, pp. 687 - 709
Copyright
Copyright © Canadian Mathematical Society 1983

References

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. Carlitz, L., Some arithmetic properties of the Olivier functions, Math. Annalen 128 (1954-1955), 412–419.Google Scholar
3. Carlitz, L., A note on Rummer's congruences, Arch. Math. 7 (1957), 441–445.Google Scholar
4. Carlitz, L., Kummer's congruences (mod 2r), Monatshefte fur Math. 63 (1959), 394–400.Google Scholar
5. Carlitz, L., Composition of sequences satisfying Kummefs congruences, Collect. Math. 11 (1959), 137–152.Google Scholar
6. Carlitz, L., Some arithmetic properties of a special sequence of integers, Can. Math. Bull. 19 (1976), 425–429.Google Scholar
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. Gessel, I. M., Some congruences for Apery numbers, J. Number Theory 14 (1982), 362–368.Google Scholar
9. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 4th ed. (Oxford University Press, 1965).Google Scholar
10. Kazandzidis, G. S., On congruences in number-theory, Bull. Soc. Math. Grèce (N. S.) 10 (1969), 35–40.Google Scholar
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. Leeming, D. J. and MacLeod, R. A., Some properties of generalized Euler numbers, Can. J. Math. 33 (1981), 606–617.Google Scholar
13. Rota, G.-C. and Sagan, B., Congruences derived from group action, Europ. J. Combinatorics 1 (1980), 67–76.Google Scholar
14. Smith, J. H., Combinatorial congruences derived from the action of Sylow subgroups of the symmetric group, preprint.Google Scholar
15. Stevens, H., Generalized Kummer congruences for products of sequences, Duke Math. J. 28 (1961), 25–38.Google Scholar
16. Stevens, H., Generalized Kummer congruences for the products of sequences. Applications, Duke Math. J. 28 (1961), 261–275.Google Scholar
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