Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials. (English) Zbl 1268.11033
The authors reveal a simple relation linking the two classes of polynomials in the title, and, therefore, they treat in detail only those associated with Bernoulli. A Fourier series developed in an earlier paper by the same authors [J. Approx. Theory 163, No. 1, 22–40 (2011; Zbl 1253.11028)] is shown to remain valid as an asymptotic expansion over compact subsets of the complex plane. This, in turn, is used to deduce explicit asymptotic estimates for the polynomials in question.
Reviewer: Tom M. Apostol (Pasadena)
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 42A10 | Trigonometric approximation |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Keywords:
Apostol-Bernoulli polynomials; Apostol-Euler polynomials; Fourier series; asymptotic estimatesCitations:
Zbl 1253.11028References:
| [1] | T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161 – 167. · Zbl 0043.07103 |
| [2] | A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Math. Comp. 80 (2011), 2219-2221. · Zbl 1267.11016 |
| [3] | Karl Dilcher, Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials, J. Approx. Theory 49 (1987), no. 4, 321 – 330. · Zbl 0609.10008 · doi:10.1016/0021-9045(87)90071-2 |
| [4] | Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. · Zbl 0052.29502 |
| [5] | 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. · Zbl 1253.11028 · doi:10.1016/j.jat.2010.02.005 |
| [6] | Qiu-Ming Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10 (2006), no. 4, 917 – 925. · Zbl 1189.11011 |
| [7] | Qiu-Ming Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp. 78 (2009), no. 268, 2193 – 2208. · Zbl 1214.11032 |
| [8] | 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. · Zbl 1099.33011 · doi:10.1016/j.camwa.2005.04.018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
