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ON POLY-EULER NUMBERS

Part of: Sequences and sets

Published online by Cambridge University Press:  03 November 2016

YASUO OHNO  and
YOSHITAKA SASAKI
YASUO OHNO
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan email ohno@math.tohoku.ac.jp
YOSHITAKA SASAKI*
Affiliation:
Liberal Arts Education Center, Osaka University of Health and Sport Sciences, Asashirodai 1-1, Kumatori-cho, Sennan-gun, Osaka 590-0496, Japan email ysasaki@ouhs.ac.jp
*
✉ysasaki@ouhs.ac.jp
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Abstract

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Poly-Euler numbers are introduced as a generalization of the Euler numbers in a manner similar to the introduction of the poly-Bernoulli numbers. In this paper, some number-theoretic properties of poly-Euler numbers, for example, explicit formulas, a Clausen–von Staudt type formula, congruence relations and duality formulas, are given together with their combinatorial properties.

Keywords

Euler numberpoly-Euler numberpoly-Bernoulli numberClausen–von Staudt theorem

MSC classification

Primary: 11B68: Bernoulli and Euler numbers and polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 1 , August 2017 , pp. 126 - 144
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was partly supported by Grant-in-Aid for Scientific Research (C) No. 23540036, 15K04774, and the second author was partly supported by Grant-in-Aid for Young Scientists (B) No. 23740036, 15K17524.

References

Arakawa, T., Ibukiyama, T. and Kaneko, M., Bernoulli Numbers and Zeta Functions (Makino Shoten Ltd, Tokyo, Japan, 2001), (in Japanese).Google Scholar
Arakawa, T. and Kaneko, M., ‘Multiple zeta values, poly-Bernoulli numbers, and related zeta functions’, Nagoya Math. J. 153 (1999), 189–209.Google Scholar
Arakawa, T. and Kaneko, M., ‘On poly-Bernoulli numbers’, Comment. Math. Univ. St. Pauli 48 (1999), 159–167.Google Scholar
Brewbaker, C., ‘Lonesum $(0,1)$ -matrices and poly-Bernoulli numbers of negative index’, Master’s thesis, Iowa State University, 2005.Google Scholar
Hardin, R. H., Sequence number A081200 in On-line Encyclopedia of Integer Sequences, http://oeis.org/A081200.Google Scholar
Kaneko, M., ‘Poly-Bernoulli numbers’, J. Théor. Nombres Bordeaux 9 (1997), 199–206.CrossRefGoogle Scholar
Katz, N., Sequence number A003462 in On-line Encyclopedia of Integer Sequences, http://oeis.org/A003462.Google Scholar
Lando, S. K., Lectures on Generating Functions, Student Mathematical Library, 27 (American Mathematical Society, 2003), 150 pages.CrossRefGoogle Scholar
Launois, S., ‘Rank t ℋ-primes in quantum matrices’, Comm. Algebra 33 (2005), 837–854.Google Scholar
Ohno, Y. and Sasaki, Y., ‘On the parity of poly-Euler numbers’, RIMS Kôkyûroku Bessatsu B32 (2012), 271–278.Google Scholar
Ohno, Y. and Sasaki, Y., ‘Periodicity on poly-Euler numbers and Vandiver type congruence for Euler numbers’, RIMS Kôkyûroku Bessatsu B44 (2013), 205–212.Google Scholar
Sánchez-Peregrino, R., ‘The Lucas congruence for Stirling numbers of the second kind’, Acta Arith. 94 (2000), 41–52.CrossRefGoogle Scholar
Sasaki, Y., ‘On generalized poly-Bernoulli numbers and related L-functions’, J. Number Theory 132 (2012), 156–170.Google Scholar