Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions

Abstract

In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known results, for example, Raabe’s multiplication formula and Hermite’s identity, are deduced as special cases.

Keywords

• Apostol–Bernoulli functions
• Bernoulli functions
• Apostol–Euler functions
• quasi-periodic Euler functions
• combinatorial identity