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.
In this paper, we present a further investigation for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. By making use of the generating function methods and summation transform ...techniques, we establish some new identities involving the products of the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. Many of the results presented here are the corresponding generalizations of some known formulas on the classical Bernoulli polynomials and the classical Genocchi polynomials.
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.
Recently, Srivastava and Pintér 1 investigated several interesting properties and relationships involving the classical as well as the generalized (or higher-order) Bernoulli and Euler polynomials. ...They also showed (among other things) that the main relationship (proven earlier by Cheon 2) can easily be put in a much more general setting. The main object of the present sequel to these earlier works is to derive several general properties and relationships involving the Apostol-Bernoulli and Apostol-Euler polynomials. Some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the (generalized) Bernoulli and (generalized) Euler polynomials. Other relationships associated with the Stirling numbers of the second kind are also considered.
The aim of this paper is to introduce and investigate several new identities related to a unification and generalization of the three families of generalized Apostol type polynomials such as the ...Apostol–Bernoulli polynomials, the Apostol–Euler polynomials and the Apostol–Genocchi polynomials. The results presented here are based upon the theory of the Umbral Calculus and the Umbral Algebra. We also introduce some operators. By using a unified generating function for these Apostol type polynomials, which was constructed recently by Özden et al. (2010) 42, we derive many new properties of these polynomials. Moreover, we give relations between these polynomials and the Stirling numbers of the first and second kind.
In this paper, we present a unified family of polynomials including not only the Apostol–Bernoulli, Euler and Genocchi polynomials, but also a general family of polynomials suggested by Özden et al. ...H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 60 (10) (2010) 2779–2787. We obtain the explicit representation of this unified family, in terms of a Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given.
The main objective of this work is to deduce some interesting algebraic relationships that connect the degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol– Genocchi polynomials and ...other families of polynomials such as the generalized Bernoulli polynomials of level m and the Genocchi polynomials. Futher, find new recurrence formulas for these three families of polynomials to study.
The main object of this paper is to investigate the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials. We first establish two relationships between the generalized Apostol–Bernoulli and ...Apostol–Euler polynomials. It can be found that many results obtained before are special cases of these two relationships. Moreover, we have a study on the sums of products of the Apostol–Bernoulli polynomials and of the Apostol–Euler polynomials.
In recent years, mathemacians (1, 3, 5, 22, 23) introduced and investigated the Fubini Apostol-type numbers and polynomials. They gave some recurrence relations explicit properties and identities for ...these polynomials. In 12, author considered unified degenerate Apostol-type Bernoulli, Euler, Genocchi and Fubini polynomials and gave some relations and identities for these polynomials. In this article, we consider a parametric unified Apostol-type Bernoulli, Euler, Genocchi and Fubini polynomials. By using the monomiality principle, we give some relations for the parametric unified Apostol-type Bernoulli, Euler, Genocchi and Fubini polynomials. Furthermore, wegive summation formula for these polynomials.
The main motivation of this paper is to investigate some derivative properties of the generating functions for the numbers Yn(λ) and the polynomials Yn(x;λ), which were recently introduced by Simsek ...30. We give functional equations and differential equations (PDEs) of these generating functions. By using these functional and differential equations, we derive not only recurrence relations, but also several other identities and relations for these numbers and polynomials. Our identities include the Apostol–Bernoulli numbers, the Apostol–Euler numbers, the Stirling numbers of the first kind, the Cauchy numbers and the Hurwitz–Lerch zeta functions. Moreover, we give hypergeometric function representation for an integral involving these numbers and polynomials. Finally, we give infinite series representations of the numbers Yn(λ), the Changhee numbers, the Daehee numbers, the Lucas numbers and the Humbert polynomials.
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