Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials
HTML articles powered by AMS MathViewer
- by Luis M. Navas, Francisco J. Ruiz and Juan L. Varona PDF
- Math. Comp. 81 (2012), 1707-1722 Request permission
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.
References
- T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161–167. MR 43843
- A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Math. Comp. 80 (2011), 2219–2221.
- Karl Dilcher, Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials, J. Approx. Theory 49 (1987), no. 4, 321–330. MR 881502, DOI 10.1016/0021-9045(87)90071-2
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
- Luis M. Navas, Francisco J. Ruiz, and Juan L. Varona, The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials, J. Approx. Theory 163 (2011), no. 1, 22–40. MR 2741217, DOI 10.1016/j.jat.2010.02.005
- Qiu-Ming Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10 (2006), no. 4, 917–925. MR 2229631, DOI 10.11650/twjm/1500403883
- Qiu-Ming Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp. 78 (2009), no. 268, 2193–2208. MR 2521285, DOI 10.1090/S0025-5718-09-02230-3
- Qiu-Ming Luo and H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51 (2006), no. 3-4, 631–642. MR 2207447, DOI 10.1016/j.camwa.2005.04.018
Additional Information
- Luis M. Navas
- Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain
- MR Author ID: 679507
- ORCID: 0000-0002-5742-8679
- Email: navas@usal.es
- Francisco J. Ruiz
- Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, Campus de la Plaza de San Francisco, 50009 Zaragoza, Spain
- Email: fjruiz@unizar.es
- Juan L. Varona
- Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, Calle Luis de Ulloa s/n, 26004 Logroño, Spain
- MR Author ID: 260232
- ORCID: 0000-0002-2023-9946
- Email: jvarona@unirioja.es
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 1707-1722
- MSC (2010): Primary 11B68; Secondary 42A10, 41A60
- DOI: https://doi.org/10.1090/S0025-5718-2012-02568-3
- MathSciNet review: 2904599