Some formulas for Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments. (English) Zbl 1189.11012

Luo, Qiu-Ming

Appl. Anal. Discrete Math. 3, No. 2, 336-346 (2009).

In [Pac. J. Math. 1, 161–167 (1951; Zbl 0043.07103)] T. M. Apostol introduced a generalization of the classical Euler polynomials as analogous definition of Apostol type for the so-called Apostol-Euler numbers and polynomials of higher order. The generalization, the Apostol-Euler polynomials \(\mathcal E_n^{(\alpha)}(x;\lambda)\), is defined by means of the following generating function: \[ \left(\frac 2{\lambda e^z+1}\right)^\alpha e^{xz} =\sum_{n=0}^\infty \mathcal E_n^{(\alpha)}(x;\lambda)\frac{z^n}{n!}\quad(| z| <| \log(-\lambda)| ), \] with: \(E_n^{(\alpha)}(x)=\mathcal E_n^{(\alpha)}(x;1)\) and \(\mathcal E_n^{(\alpha)}(\lambda):=2^n\mathcal{E}_n^{(\alpha)}(\alpha/2;\lambda)\); \(\mathcal E_n(x;\lambda)=\mathcal{E}_n^{(1)}(x;\lambda)\) and \(\mathcal E_n(\lambda):=2^n\mathcal{E}_n(\alpha/2;\lambda)\), where \(\mathcal E_n(\lambda)\), \(\mathcal{E}_n^{(\alpha)}(\lambda)\) and \(\mathcal E_n(x;\lambda)\) denote the so-called Apostol-Euler numbers, Apostol-Euler numbers of order \(\alpha\) and Apostol-Euler polynomials, respectively. In a similar manner the Apostol-Bernoulli polynomials of order \(\alpha\), a generalization of the classical Bernoulli polynomials, is introduced. The author derives some relationships between these polynomials and the generalized Hurwitz–Lerch zeta-function. He also gives an explicit series representations for the polynomials involving the Hurwitz zeta-function and the Riemann zeta-function.

Reviewer: Dragan Stankov (Beograd)

MSC:

11B68	Bernoulli and Euler numbers and polynomials
11M35	Hurwitz and Lerch zeta functions
11B73	Bell and Stirling numbers
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

Bernoulli polynomials; Apostol-Bernoulli polynomials; Gaussian hypergeometric function; Stirling numbers of the second kind; Hurwitz zeta function; Lerch functional equation

Citations:

Zbl 0043.07103