Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials |
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Authors: | Qiu-Ming Luo |
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Affiliation: | a Department of Mathematics, Jiaozuo University, Jiaozuo City, Henan 454003, People's Republic of China b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada |
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Abstract: | The main object of this paper is to give analogous definitions of Apostol type (see T.M. Apostol, On the Lerch Zeta function, Pacific J. Math. 1 (1951) 161-167] and H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77-84]) for the so-called Apostol-Bernoulli numbers and polynomials of higher order. We establish their elementary properties, derive several explicit representations for them in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) Zeta function, and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical Bernoulli numbers and polynomials of higher order. |
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Keywords: | Bernoulli polynomials Apostol-Bernoulli polynomials Apostol-Bernoulli polynomials of higher order Apostol-Euler polynomials Apostol-Euler polynomials of higher order Gaussian hypergeometric function Stirling numbers of the second kind Hurwitz (or generalized) Zeta function Hurwitz-Lerch and Lipschitz-Lerch Zeta functions Lerch's functional equation |
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