Apostol-Bernoulli Polynomials Research Articles

Some properties of the generalized Apostol-type polynomials

In this paper, we study some properties of the generalized Apostol-type polynomials (see (Luo and Srivastava in Appl. Math. Comput. 217:5702-5728, 2011)), including the recurrence relations, the differential equations and some other connected problems, which extend some known results. We also deduce some properties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli polynomials, and Apostol-Genocchi polynomials of high order.MSC:11B68, 33C65.

An extension of generalized Apostol-Euler polynomials

Abstract Recently, Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials (see Tremblay in Appl. Math. Let. 24:1888-1893, 2011). In this paper, we introduce and investigate an extension of the generalized Apostol-Euler polynomials. We state some properties for these polynomials and obtain some relationships between the polynomials and Apostol-Bernoulli polynomials, Stirling numbers of the second kind, Jacobi polynomials, Laguerre polynomials, Hermite polynomials and generalized Bernoulli polynomials. MSC:11B68, 11B73, 33C45.

A FURTHER GENERALIZATION OF APOSTOL-BERNOULLI POLYNOMIALS AND RELATED POLYNOMIALS

The purpose of this paper is to introduce and investigate two new classes of generalized Bernoulli and Apostol-Bernoulli polynomials based on the definition given recently by the authors [29]. In particular, we obtain a new addition formula for the new class of the generalized Bernoulli polynomials. We also give an extension and some analogues of the Srivastava-Pint<TEX>$\acute{e}$</TEX>r addition theorem [28] for both classes. Finally, by making use of the new adition formula, we exhibit several interesting relationships between generalized Bernoulli polynomials and other polynomials or special functions.

Higher recurrences for Apostol-Bernoulli-Euler numbers

We present explicit formulas for sums of products of Apostol-Bernoulli and Apostol-Euler numbers of the form $$\sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)B_{m_1 } (q) \cdots B_{m_N } (q),} \sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)E_{m_1 } (q) \cdots E_{m_N } (q),}$$ where N and n are positive integers, B m (q) n stand for the Apostol-Bernoulli numbers, E m (q) for the Apostol-Euler numbers, and $$\left( {\begin{array}{*{20}c} n \\ {m_1 , \cdots ,m_N } \\ \end{array} } \right) = \frac{{n!}} {{m_1 ! \cdots m_N !}}.$$ Our formulas involve Stirling numbers of the first kind. We also derive results for Apostol-Bernoulli and Apostol-Euler polynomials. As an application, for q = 1 we recover results of Dilcher, and our paper can be regarded as a q-extension of that of Dilcher.

Sums of products of Apostol–Bernoulli numbers

By expressing the sums of products of the Apostol–Bernoulli polynomials in terms of the special values of multiple Hurwitz–Lerch zeta functions at non-positive integers, we obtain the sums of products identity for the Apostol–Bernoulli numbers which is an analogue of the classical sums of products identity for Bernoulli numbers dating back to Euler.

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $\mathcal{B}_{n}(x;\lambda)$ in detail. The starting point is their Fourier series on $[0,1]$ which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials $\mathcal{E}_{n}(x;\lambda)$ via a simple relation linking them to the Apostol-Bernoulli polynomials.

Some Formulae of Products of the Apostol‐Bernoulli and Apostol‐Euler Polynomials

Some formulae of products of the Apostol‐Bernoulli and Apostol‐Euler polynomials are established by applying the generating function methods and some summation transform techniques, and various known results are derived as special cases.

Some series identities involving the generalized Apostol type and related polynomials

A unification (and generalization) of various Apostol type polynomials was introduced and investigated recently by Luo and Srivastava [Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728]. In this paper, we prove several symmetry identities for these generalized Apostol type polynomials by using their generating functions. As special cases and consequences of our results, we obtain the corresponding symmetry identities for the Apostol–Euler polynomials of higher order, the Apostol–Bernoulli polynomials of higher order and the Apostol–Genocchi polynomials of higher order, and also for another family of generalized Apostol type polynomials which were investigated systematically by Ozden 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 (2010) 2779–2787]. We also derive several relations between the Apostol type polynomials, the generalized sum of integer powers and the generalized alternating sum. It is shown how each of these results would extend the corresponding known identities.

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.

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.

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].

On the apostol-bernoulli polynomials

Abstract In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications.