## Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions.(English)Zbl 1184.11005

The reviewer and H. M. Srivastava defined the Apostol-Bernoulli polynomials of higher order as follows [J. Math. Anal. Appl. 308, No. 1, 290–302 (2005; Zbl 1076.33006)]: $\left(\frac{z}{\lambda e^{z}-1}\right)^{\alpha}\;e^{xz}=\sum_{n=0}^{\infty}\mathcal{B}_{n}^{(\alpha)}(x;\lambda)\;\frac{z^{n}}{n!}$ ($$|z| <2\pi$$ when $$\lambda=1$$; $$|z| <| \log \lambda |$$ when $$\lambda \neq 1$$) with, of course, \begin{aligned} B_{n}^{(\alpha)}(x)=\mathcal{B}_{n}^{(\alpha)}(x;1)&\text{ and } \mathcal{B}_{n}^{(\alpha)}(\lambda):=\mathcal{B}_{n}^{(\alpha)}(0;\lambda),\\ \mathcal{B}_{n}(x;\lambda):=\mathcal{B}_{n}^{(1)}(x;\lambda)&\text{ and } \mathcal{B}_{n}(\lambda):=\mathcal{B}_{n}^{(1)}(\lambda), \end{aligned} where $$\mathcal{B}_{n}(\lambda)$$, $$\mathcal{B}_{n}^{(\alpha)}(\lambda)$$ and $$\mathcal{B}_{n}(x;\lambda)$$ denote the so-called Apostol-Bernoulli numbers, Apostol-Bernoulli numbers of order $$\alpha$$, and Apostol-Bernoulli polynomials, respectively.
The reviewer also defined the Apostol-Euler polynomials of higher order as follows [Taiwanese J. Math. 10, No. 4, 917–925 (2006; Zbl 1189.11011)]: $\left(\frac{2}{\lambda e^{z}+1}\right)^{\alpha }\;e^{xz}=\sum_{n=0}^{\infty}\mathcal{E}_{n}^{(\alpha)}(x;\lambda)\frac{z^{n}}{n!} \qquad (| z | <| \log (-\lambda)|),$ with, of course, \begin{aligned} E_{n}^{(\alpha)}(x)=\mathcal{E}_{n}^{(\alpha)}(x;1)&\text{ and } \mathcal{E}_{n}^{(\alpha)}(\lambda):=2^n\mathcal{E}_{n}^{(\alpha)}\left(\frac{\alpha}{2};\lambda\right),\\ \mathcal{E}_{n}(x;\lambda):=\mathcal{E}_{n}^{(1)}(x;\lambda)&\text{ and }\mathcal{E}_{n}(\lambda):=\mathcal{E}_{n}^{(1)}(\lambda), \end{aligned} 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.
A family of the Hurwitz-Lerch zeta-functions $$\Phi_{\mu,\nu}^{(\rho,\sigma)} (z,s,a)$$ defined by [S.-D. Lin, H. M. Srivastava and P.-Y. Wang, Integral Transforms Spec. Funct. 17, No. 11, 817–827 (2006; Zbl 1172.11026)]. $\Phi_{\mu,\nu}^{(\rho,\sigma)} (z,s,a):=\sum_{n=0}^{\infty}\frac{(\mu)_{\rho{n}}}{(\nu)_{\sigma{n}}}\frac{z^{n}}{(n+a)^{s}}$
($$\mu \in \mathbb{C}$$; $$a, \nu \in \mathbb{C}\setminus \mathbb{Z}_{0}^{-}$$; $$\rho, \sigma \in \mathbb{R}^+$$; $$\rho < \sigma$$ when $$s, z \in \mathbb{C}$$; $$\rho=\sigma$$ and $$s \in \mathbb{C}$$ when $$|z |<1$$; $$\rho=\sigma$$ and $$\text{Re}(s-\mu+\nu)>1$$ when $$|z |= 1$$) contains, as its special cases, not only the Hurwitz-Lerch zeta-function $\Phi_{\nu,\nu}^{(\sigma,\sigma)} (z,s,a)=\Phi_{\mu,\nu}^{(0,0)} (z,s,a)=\Phi (z,s,a)=\sum_{n=0}^{\infty }\frac{z^{n}}{(n+a)^{s}}$ but also the following generalized Hurwitz zeta-functions introduced and studied earlier by S. P. Goyal and R. K. Laddha [see Gaṇita Sandesh 11, 99–108 (1997)] $\Phi_{\mu,1}^{(1,1)} (z,s,a)=\Phi_{\mu} (z,s,a):=\sum_{n=0}^{\infty}\frac{(\mu)_{n}}{n!}\frac{z^{n}}{(n+a)^{s}}.$
In this paper, the authors further investigate the generalized Apostol-Bernoulli polynomials of higher order. First they find an important relationship between the Apostol-Bernoulli polynomials of higher order and the generalized Hurwitz-Lerch zeta-function as follows: $\mathcal{B}_{n}^{(l)}(a;\lambda)= (-n)_l \Phi_l(\lambda,l-n,a)\qquad (n, l \in \mathbb{N}; \;n \geqq l; \;|\lambda|<1; \;a \in \mathbb{C} \setminus \mathbb{Z}_0^{-}),$ In particular, for $$l=1$$ $\mathcal{B}_{n}(a;\lambda)= -n \Phi(\lambda,1-n,a)\qquad (n \in \mathbb{N}; \;|\lambda| \leqq 1; \;a \in \mathbb{C} \setminus \mathbb{Z}_0^{-}).$ are just an extension of the T. M. Apostol’s formula [Pac. J. Math. 1, 161–167 (1951; Zbl 0043.07103)]. $\phi (\xi ,a,1-n)=\Phi (e^{2\pi i \xi} ,1-n,a)=-\frac{\mathcal{B}_{n}(a;e^{2\pi i\xi })}{n},\qquad (n\in \mathbb{N}).$
Next they obtain some formulas of the generalized Apostol-Bernoulli polynomials at rational arguments in terms of the Hurwitz zeta-function.
Reviewer’s remark: Srivastava used Apostol’s formula $\phi (\xi ,a,1-n)=\Phi (e^{2\pi i \xi} ,1-n,a)=-\frac{\mathcal{B}_{n}(a;e^{2\pi i\xi })}{n},\qquad (n\in \mathbb{N})$ and Lerch’s functional equation $\begin{split} \phi (\xi ,a,1-s)=\;\frac{\Gamma (s)}{(2\pi)^{s}}\left\{ \exp \left[\left(\frac{1}{2}s-2a\xi\right)\pi i\right] \phi (-a,\xi ,s)\right. \\ \left. +\exp \left[\left(-\frac{1}{2}s+2a(1-\xi)\right)\pi i\right]\phi (a,1-\xi ,s)\right \}, \end{split}$ ($$s\in \mathbb{C}$$; $$0<\xi <1$$) to derive an elegant formula of Apostol-Bernoulli polynomials at rational argument [H. M. Srivastava, Math. Proc. Camb. Philos. Soc. 129, No. 1, 77–84 (2000; Zbl 0978.11004)]:
$\begin{split} \mathcal{B}_{n}\left(\frac{p}{q};e^{2\pi i\xi}\right)= -\frac{n!}{(2q\pi)^{n}}\Bigg \{ \sum_{j=1}^{q}\zeta \left(n,\frac{\xi +j-1}{q}\right)\exp \left[\left(\frac{n}{2}-\frac{2(\xi +j-1)p}{q}\right)\pi i\right]\\ +\sum_{j=1}^{q}\zeta \left(n,\frac{j-\xi }{q}\right)\exp \left[\left(-\frac{n}{2}+\frac{2(j-\xi)p}{q}\right)\pi i\right] \Bigg \} , \end{split}$
($$n \in \mathbb{N} \setminus \{1\}$$; $$q \in \mathbb{N}$$; $$p \in \mathbb{Z}$$; $$\xi \in \mathbb{R}).$$
Recently, the reviewer further obtained the following relationships between the Apostol-Euler polynomials of higher order and the generalized Hurwitz zeta function: $\mathcal{E}_{n}^{(\alpha)}(a;\lambda)=2^\alpha \Phi_{\alpha }(-\lambda,-n,a)\qquad (n \in \mathbb{N}; \;|\lambda| \leqq 1; \;\alpha \in \mathbb{C}; \;a \in \mathbb{C} \setminus \mathbb{Z}_0^{-})$ and $\mathcal{E}_{n}(a;\lambda)= 2 \Phi(-\lambda,-n,a)\qquad (n \in \mathbb{N}; \;|\lambda| \leqq 1; \;a \in \mathbb{C} \setminus \mathbb{Z}_0^{-}).$ and derived the formulas of the generalized Apostol-Euler polynomials at rational arguments as follows: $\begin{split} \mathcal{E}_{n}\left(\frac{p}{q};e^{2\pi i \xi}\right)\\ = \frac{2 \cdot n!}{(2q \pi)^{n+1}}\Bigg \{\sum_{j=1}^{q}\zeta \left(n+1,\frac{2\xi +2j-1}{2q}\right) \exp \left[\left(\frac{n+1}{2}-\frac{(2\xi +2j-1)p}{q}\right)\pi i \right]\\ + \sum_{j=1}^{q}\zeta \left(n+1,\frac{2j-2\xi-1}{2q}\right)\exp \left[\left(-\frac{n+1}{2}+\frac{(2j-2\xi-1)p}{q}\right)\pi i \right]\Bigg \}, \end{split}$ $$n, q \in \mathbb{N}$$; $$p \in \mathbb{Z}$$; $$\xi \in \mathbb{R}$$.

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials 11B65 Binomial coefficients; factorials; $$q$$-identities 11B73 Bell and Stirling numbers 11M35 Hurwitz and Lerch zeta functions 33C05 Classical hypergeometric functions, $${}_2F_1$$
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