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Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

Authors: Luis M. Navas, Francisco J. Ruiz and Juan L. Varona
Journal: Math. Comp. 81 (2012), 1707-1722
MSC (2010): Primary 11B68; Secondary 42A10, 41A60
Published electronically: January 12, 2012
MathSciNet review: 2904599
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Abstract: We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $ \mathcal {B}_{n}(x;\lambda )$ in detail. The starting point is their Fourier series on $ [0,1]$ which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases.

These results are transferred to the Apostol-Euler polynomials $ \mathcal {E}_{n}(x;\lambda )$ via a simple relation linking them to the Apostol-Bernoulli polynomials.

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Additional Information

Luis M. Navas
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain

Francisco J. Ruiz
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, Campus de la Plaza de San Francisco, 50009 Zaragoza, Spain

Juan L. Varona
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, Calle Luis de Ulloa s/n, 26004 Logroño, Spain

Keywords: Apostol-Bernoulli polynomials, Apostol-Euler polynomials, Fourier series, asymptotic estimates
Received by editor(s): February 7, 2011
Received by editor(s) in revised form: April 27, 2011
Published electronically: January 12, 2012
Additional Notes: Research of the second and third authors supported by grant MTM2009-12740-C03-03 of the DGI
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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