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Mathematics of Computation

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Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials

Author: Qiu-Ming Luo
Journal: Math. Comp. 78 (2009), 2193-2208
MSC (2000): Primary 11B68; Secondary 42A16, 11M35
Published electronically: June 12, 2009
MathSciNet review: 2521285
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Abstract: We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known results.

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Additional Information

Qiu-Ming Luo
Affiliation: Department of Mathematics, East China Normal University, Shanghai 200241, People’s Republic of China –and– Department of Mathematics, Jiaozuo University, Henan Jiaozuo 454003, People’s Republic of China

Keywords: Lipschitz summation formula, Fourier expansion, integral representation, Apostol-Bernoulli and Apostol-Euler polynomials and numbers, Bernoulli and Euler polynomials and numbers, Hurwitz Zeta function, Lerch's functional equation, rational arguments
Received by editor(s): June 3, 2008
Received by editor(s) in revised form: September 26, 2008
Published electronically: June 12, 2009
Additional Notes: The author expresses his sincere gratitude to the referee for valuable suggestions and comments. The author thanks Professor Chi-Wang Shu who helped with the submission of this manuscript to the Web submission system of the AMS.
The present investigation was supported in part by the PCSIRT Project of the Ministry of Education of China under Grant #IRT0621, Innovation Program of Shanghai Municipal Education Committee of China under Grant #08ZZ24 and Henan Innovation Project For University Prominent Research Talents of China under Grant #2007KYCX0021.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.