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Generalized Euler Number Sequences: Asymptotic Estimates and Congruences

Published online by Cambridge University Press:  20 November 2018

D. J. Leeming  and
R. A. Macleod
D. J. Leeming
Affiliation:
University of Victoria, Victoria, British Columbia
R. A. Macleod
Affiliation:
University of Victoria, Victoria, British Columbia
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We define (as in [7]) integer sequences one for each positive integer k ≧ 2, by

1.1

where are the kth roots of unity and (E(k))n is replaced by after multiplying out. We note that (1.1) implies , n ≠ 0 (mod k).

In [7], we considered some special properties of these number sequences, proved several congruences and conjectured several others. This paper is a continuation of the work presented in [7].

In Section 2 we demonstrate the asymptotic rate of growth of the numbers by showing that

In Section 3 we present a large number of congruences (modulo 2048), some of which are proved or can be proved by the techniques presented herein, and other congruences which appear to be true on the basis of numerical evidence.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 3 , 01 June 1983 , pp. 526 - 546
Copyright
Copyright © Canadian Mathematical Society 1983

References

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