Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind

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 investig...

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Vydáno v:	Applied mathematics and computation Ročník 217; číslo 12; s. 5702 - 5728
Hlavní autoři:	Luo, Qiu-Ming, Srivastava, H.M.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier Inc 15.02.2011 Elsevier
Témata:	Algebra Analogue Apostol–Bernoulli polynomials and Apostol–Euler polynomials of higher order Apostol–Genocchi numbers and Apostol–Genocchi polynomials Apostol–Genocchi numbers and Apostol–Genocchi polynomials of higher order Computation Difference and functional equations, recurrence relations Error correction Exact sciences and technology Functions of a complex variable Gaussian Genocchi numbers and Genocchi polynomials of higher order Hurwitz (or generalized), Hurwitz–Lerch and Lipschitz–Lerch zeta functions Hypergeometric functions Lerch’s functional equation Mathematical analysis Mathematical models Mathematics Number theory Numerical analysis Numerical analysis. Scientific computation Representations Sciences and techniques of general use Special functions Srivastava’s formula and Gaussian hypergeometric function Stirling numbers and the λ-Stirling numbers of the second kind Lerch’s functional equation Hurwitz (or generalized), Hurwitz–Lerch and Lipschitz–Lerch zeta functions Srivastava’s formula and Gaussian hypergeometric function Genocchi numbers and Genocchi polynomials of higher order Stirling numbers and the λ-Stirling numbers of the second kind Apostol–Genocchi numbers and Apostol–Genocchi polynomials of higher order Apostol–Bernoulli polynomials and Apostol–Euler polynomials of higher order Apostol–Genocchi numbers and Apostol–Genocchi polynomials Transcendental equation Apostol-Bernoulli polynomials and Error estimation Euler polynomial Srivastava's formula and Gaussian hypergeometric function Recurrence relation Functional equation Hurwitz (or generalized), Hurwitz-Lerch and Lipschitz-Lerch zeta functions Difference equation Apostol-Genocchi numbers and Apostol-Genocchi polynomials Lerch's functional equation Numerical analysis Applied mathematics Apostol―Genocchi numbers and Apostol-Genocchi polynomials of higher order Hypergeometric function Zeta function Lipschitz function Apostol―Euler polynomials of higher order
ISSN:	0096-3003, 1873-5649
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Shrnutí:	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.
Bibliografie:	ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0096-3003 1873-5649
DOI:	10.1016/j.amc.2010.12.048