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

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.

### 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.)

Zbl 0043.07103
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