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Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind
Authors:Qiu-Ming Luo
Institution:a Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing 401331, People’s Republic of China
b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
Abstract:Recently, the authors introduced some generalizations of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials (see Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290-302] and Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917-925]). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol-Genocchi polynomials of higher order. For these generalized Apostol-Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631-642] and pose two open problems on the subject of our investigation.
Keywords:Genocchi numbers and Genocchi polynomials of higher order  Apostol-Genocchi numbers and Apostol-Genocchi polynomials  Apostol-Genocchi numbers and Apostol-Genocchi polynomials of higher order  Apostol-Bernoulli polynomials and Apostol-Euler polynomials of higher order  Srivastava&rsquo  s formula and Gaussian hypergeometric function  Hurwitz (or generalized)  Hurwitz-Lerch and Lipschitz-Lerch zeta functions  Lerch&rsquo  s functional equation  Stirling numbers and the λ-Stirling numbers of the second kind
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