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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Möbius inversion formulae for Apostol-Bernoulli type polynomials and numbers


Authors: A. Bayad and J. Chikhi
Journal: Math. Comp. 82 (2013), 2327-2332
MSC (2010): Primary 11B68, 11A25, 11B73, 42A16, 41A58
Published electronically: April 24, 2013
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Abstract: In this paper, we establish Möbius inversion formulae for the Fourier expansions of the Apostol-Bernoulli, Apostol-Euler and Apostol-
Genocchi polynomials. As an application, by specializing our formulae at some special values we obtain interesting number-theoritical relations. We derive explicit formulae for Apostol-Bernoulli numbers. These formulae involve Stirling numbers of the second kind and powers of cotangent. Our proofs are very simple.


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Additional Information

A. Bayad
Affiliation: Département de mathématiques, Université d’Evry Val d’Essonne, Bâtiment I.B.G.B.I., 3ème étage, 23 Bd. de France, 91037 Evry Cedex, France
Email: abayad@maths.univ-evry.fr

J. Chikhi
Affiliation: Département de mathématiques, Université d’Evry Val d’Essonne, Bâtiment I.B.G.B.I., 3ème étage, 23 Bd. de France, 91037 Evry Cedex, France
Email: jchikhi@univ-evry.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02699-3
PII: S 0025-5718(2013)02699-3
Keywords: Apostol-Bernoulli numbers and polynomials, Apostol-Euler polynomials, Apostol-Genocchi polynomials, Stirling numbers, Fourier series, M\"obius function, M\"obius inversion
Received by editor(s): November 20, 2013
Received by editor(s) in revised form: January 25, 2012
Published electronically: April 24, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.