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

Abstract

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.  2005 Elsevier Inc. All rights reserved.

Unified laguerre-based poly-Apostol-type polynomials

“…Luo (2009), Luo and Srivastava (2005) and Srivastava (2011) introduced the Apostol-Bernoulli, Apostol-Euler and ApostolGenocchi polynomials and proved some theorems and relations for these polynomials. Kurt (2016aKurt ( , 2016b introduced the unified family of generalized Apostol-type polynomials and gave some symmetry identities and recurrences relations for these polynomials.…”

A New Class of Laguerre-based Generalized Apostol Polynomials

“…where E n (x) denotes the classical Euler polynomials (see from example [5], [6], [7], [9], [12], [31]- [33], [35]- [40]; see also the reference cited in each of these earlier works).…”

IDENTITIES OF SYMMETRY FOR THE HIGHER ORDER q-BERNOULLI POLYNOMIALS

“…Luo and Srivastava [21] generalized this definition to obtain the generalized ApostolBernoulli and Euler polynomials, and they also studied them systematically. Recently, Luo [20], Bayad [2], Navas, Francisco and Varona [23] investigated Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials.…”