Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials

Abstract

Abstract. We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known results.

Algebraic properties and Fourier expansions of two-dimensional Apostol–Bernoulli and Apostol–Euler polynomials

“…The results obtained generalize those in [2,6,8]. These are interesting results both from the theoretical and computational point of view, as they allow the use of Fourier approximation to both calculate and derive further relations between the various polynomial families involved.…”

Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

“…(1.6) Remark 1.4. Luo's proof [4], for Theorems 1.1 and 1.2, uses the Lipschitz summation formula [2] which is not easy to understand. In this paper we propose a very simple proof.…”

Some new identities for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials

“…Luo [29,31] derived some multiplication formulas for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials. Further, Luo [28,30] gave the Fourier expansions for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials by applying the Lipschitz summation formula and obtained some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz (or generalized) zeta function. Tremblay et al [47] investigated a new class of generalized Apostol-Bernoulli polynomials and obtained a generalization of the Srivastava-Pintér addition theorem (see [46]).…”