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Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials. (English) Zbl 1214.11032
The Lipschitz summation formula is used to obtain Fourier expansions and integral representations of the polynomials in the title. Explicit formulas at rational arguments for these polynomials are given in terms of the Hurwitz zeta function. Integral representations for the classical Bernoulli and Euler polynomials are also described.

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11M35 Hurwitz and Lerch zeta functions
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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[1] Milton Abramowitz and Irene A. Stegun , Handbook of mathematical functions with formulas, graphs, and mathematical tables, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York; John Wiley & Sons, Inc., New York, 1984. Reprint of the 1972 edition; Selected Government Publications. Irene A. Stegun , Pocketbook of mathematical functions, Verlag Harri Deutsch, Thun, 1984. Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun; Material selected by Michael Danos and Johann Rafelski.
[2] T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161 – 167. · Zbl 0043.07103
[3] L. Carlitz, Multiplication formulas for products of Bernoulli and Euler polynomials, Pacific J. Math. 9 (1959), 661 – 666. · Zbl 0089.28002
[4] Mehmet Cenkci and Mümün Can, Some results on \?-analogue of the Lerch zeta function, Adv. Stud. Contemp. Math. (Kyungshang) 12 (2006), no. 2, 213 – 223. · Zbl 1098.11016
[5] Junesang Choi, P. J. Anderson, and H. M. Srivastava, Some \?-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order \?, and the multiple Hurwitz zeta function, Appl. Math. Comput. 199 (2008), no. 2, 723 – 737. · Zbl 1146.33001
[6] Djurdje Cvijović and Jacek Klinowski, New formulae for the Bernoulli and Euler polynomials at rational arguments, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1527 – 1535. · Zbl 0827.11012
[7] Djurdje Cvijović, The Haruki-Rassias and related integral representations of the Bernoulli and Euler polynomials, J. Math. Anal. Appl. 337 (2008), no. 1, 169 – 173. · Zbl 1215.11015
[8] Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. II, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1953 original. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1955 original. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. II, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1953 original. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1955 original. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. Based, in part, on notes left by Harry Bateman.
[9] Mridula Garg, Kumkum Jain, and H. M. Srivastava, Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions, Integral Transforms Spec. Funct. 17 (2006), no. 11, 803 – 815. · Zbl 1184.11005
[10] Hiroshi Haruki and Themistocles M. Rassias, New integral representations for Bernoulli and Euler polynomials, J. Math. Anal. Appl. 175 (1993), no. 1, 81 – 90. · Zbl 0776.11009
[11] D. H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly 95 (1988), no. 10, 905 – 911. · Zbl 0663.10009
[12] R. Lipschitz, Untersuchung der Eigenschaften einer Gattung von unendlichen Reihen, J. Reine und Angew. Math. CV (1889), 127-156.
[13] Qiu-Ming Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308 (2005), no. 1, 290 – 302. · Zbl 1076.33006
[14] 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
[15] 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
[16] Q.-M. Luo, An explicit relationship between the generalized Apostol-Bernoulli and Apostol-Euler polynomials associated with \( \lambda-\)Stirling numbers of the second kind, Houston J. Math., accepted in press. · Zbl 1221.11062
[17] Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transforms Spec. Funct., accepted in press. · Zbl 1237.11009
[18] Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0039.07202
[19] Paul C. Pasles and Wladimir de Azevedo Pribitkin, A generalization of the Lipschitz summation formula and some applications, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3177 – 3184. · Zbl 1125.11322
[20] H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 1, 77 – 84. · Zbl 0978.11004
[21] H. M. Srivastava and Junesang Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 1014.33001
[22] Weiping Wang, Cangzhi Jia, and Tianming Wang, Some results on the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 55 (2008), no. 6, 1322 – 1332. · Zbl 1151.11012
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