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Some Properties of Generalized Euler Numbers

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 infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by

(1.1)

where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is

(1.2)

Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).

Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 3 , 01 June 1981 , pp. 606 - 617
Copyright
Copyright © Canadian Mathematical Society 1981

References

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