Apostol Bernoulli Research Articles

Unified Apostol–Bernoulli, Euler and Genocchi polynomials

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.

A new class of generalized Apostol–Bernoulli polynomials and some analogues of the Srivastava–Pintér addition theorem

The main purpose of this paper is to introduce and investigate a new class of generalized Apostol–Bernoulli polynomials based on a definition given by Natalini and Bernardini (2003) [22] for the generalized Bernoulli polynomials. We obtain a generalization of the Srivastava–Pintér addition theorem (Srivastava and Pintér (2004) [23]). We also give a list of expressions involving special functions that could be used to obtain some analogues of the Srivastava–Pintér addition theorem. Finally, we give an analogue featuring the new class of generalized Apostol–Bernoulli polynomials.

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 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.

Some results on power sums and Apostol-type polynomials

In this paper, we study the Apostol–Bernoulli polynomials, the Apostol–Euler polynomials and the power sums with respect to λ. As a result, various identities are established, including the multiplication formulae of the Apostol-type polynomials, some symmetric identities and some convolution identities.

Padé approximation and Apostol–Bernoulli and Apostol–Euler polynomials

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol–Bernoulli and between Apostol–Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.

The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order

This article obtains the multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order and deduces some explicit recursive formulas. The λ-multiple power sum and λ-multiple alternating sum that are clearly evaluated associated with the Apostol–Bernoulli and Apostol–Euler polynomials of higher order, respectively, are also introduced. Some earlier results of Carlitz and Howard can be deduced.

Several identities for the generalized Apostol–Bernoulli polynomials

The purpose of this paper is to give several symmetric identities on the generalized Apostol–Bernoulli polynomials by applying the generating functions. These results extend some known identities.

An explicit formula for generalized potential polynomials and its applications

We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol–Bernoulli, Apostol–Euler polynomials and the polynomials A n ( z ) ( τ ) involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers A n ( z ) , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.

Some q-extensions of the Apostol–Bernoulli and the Apostol–Euler polynomials of order n, and the multiple Hurwitz zeta function

In this paper, we first investigate several further interesting properties of the multiple Hurwitz–Lerch Zeta function Φ n ( z, s, a) which was introduced recently by Choi et al. [J. Choi, D.S. Jang, H.M. Srivastava, A generalization of the Hurwitz–Lerch Zeta function, Integral Transform. Spec. Funct., 19 (2008)]. We then introduce and investigate some q-extensions of the multiple Hurwitz–Lerch Zeta function Φ n ( z, s, a), the Apostol–Bernoulli polynomials B k ( n ) ( x ; λ ) of order n, and the Apostol–Euler polynomials E k ( n ) ( x ; λ ) of order n. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.

Some results on the Apostol–Bernoulli and Apostol–Euler polynomials

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.

Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions

The main object of this paper is to further investigate the generalized Apostol–Bernoulli polynomials of higher order, which were introduced and studied recently by Luo and Srivastava [2005, Journal of Mathematical Analysis and Applications, 308, 290–302; 2006, Computers and Mathematics with Applications, 51, 631–642]. Here, we first derive an explicit representation of these generalized Apostol–Bernoulli polynomials of higher order in terms of a generalization of the Hurwitz–Lerch Zeta function and then proceed to establish a functional relationship between the generalized Apostol–Bernoulli polynomials of rational arguments and the Hurwitz (or generalized) Zeta function. Our results would provide extensions of those given earlier by (for example) Apostol [1951, Pacific Journal of Mathematics, 1, 161–167] and Srivastava [2000, Mathematical Proceedings of the Cambridge Philosophical Society, 129, 77–84].

Some generalizations of the Apostol–Bernoulli and Apostol–Euler 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.