Generalized Apostol-Bernoulli Polynomials Research Articles

On the perturbative expansion of exact bi-local correlators in JT gravity

We study the perturbative series associated to bi-local correlators in Jackiw-Teitelboim (JT) gravity, for positive weight λ of the matter CFT operators. Starting from the known exact expression, derived by CFT and gauge theoretical methods, we reproduce the Schwarzian semiclassical expansion beyond leading order. The computation is done for arbitrary temperature and finite boundary distances, in the case of disk and trumpet topologies. A formula presenting the perturbative result (for λ ∈ ℕ/2) at any given order in terms of generalized Apostol-Bernoulli polynomials is also obtained. The limit of zero temperature is then considered, obtaining a compact expression that allows to discuss the asymptotic behaviour of the perturbative series. Finally we highlight the possibility to express the exact result as particular combinations of Mordell integrals.

A construction of p-adic Hurwitz–Lerch L-function

We derive the existence of p-adic Hurwitz–Lerch L-function by means of a method provided by Washington. This function is a generalization of the one-variable p-adic L-function of Kubota and Leopoldt, and two-variable p-adic L-function of Fox. We also deduce divisibility properties of generalized Apostol–Bernoulli polynomials, in particular establish Kummer-type congruences for them.

Some Relationships between the Generalized Apostol-Bernoulli and Apostol-Euler Polynomials

The main objective of this paper is to introduce and investigate two new classes of generalized Apostol-Bernoulli polynomials B<SUB>n</SUB><SUP>[</SUP><SUP>m-1,</SUP>α<SUP>]</SUP>(x;c,α;λ) and Apostol-Euler polynomials ε<SUB>n</SUB><SUP>[m-1,</SUP>α<SUP>]</SUP>(x;c,α;λ). In particular, we obtain addition formula for the new class of the generalized Apostol-Bernoulli polynomials. We also give some recurrence relations and Raabe relations for these polynomials.

Some Properties of the Generalized Apostol Type Hermite-Based Polynomials

Abstract. In this paper, we study some properties of the generalized Apostol type Hermite-based polynomials. which extend some known results. We also deduce some properties of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials of high order. Numerous properties of these polynomials and some relationships between F (α) n (x; λ; μ, ν, c) and HF (α) n (x, y; λ; μ, ν, c) are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

Srivastava-Pintér theorems for 2D-Appell polynomials and their applications

Recently, Srivastava and Pinter proved addition theorems for the generalized Bernoulli and Euler polynomials. Luo and Srivastava obtained the anologous results for the generalized Apostol–Bernoulli polynomials and the generalized Apostol–Euler polynomials. Finally, Tremblay et al. gave analogues of the Srivastava–Pinter addition theorem for general family of Bernoulli polynomials. In this paper, we obtain Srivastava–Pinter type theorems for 2D-Appell Polynomials. We also give the representation of 2D-Appell Polynomials in terms of the Stirling numbers of the second kind and 1D-Appell polynomials. Furthermore, we introduce the unified 2D-Apostol polynomials. In particular, we obtain some relations between that family of polynomials and the generalized Hurwitz–Lerch zeta function as well as the Gauss hypergeometric function. Finally, we present some applications of Srivastava–Pinter type theorems for 2D-Appell Polynomials. Copyright © 2013 John Wiley & Sons, Ltd.

Hecke operators type and generalized Apostol-Bernoulli polynomials

Abstract In this paper, we construct some Hecke-type operators acting on the complex polynomials space, and we prove their commutativity. By means of this commutativity, we find a new approach to establish the generating function of the Apostol-Bernoulli type polynomials which are eigenfunctions of these Hecke-type operators. Moreover, we derive many useful identities related to these operators and polynomials. MSC:11M35, 30B40, 30B50.

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.

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.

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 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].