## 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, $${}_2F_1$$
Full Text:

### References:

  Nörlund N. E., Vorlesungen über Differentzenrechnung (1954)  Luke Y. L., The Special Functions and Their Approximations (1969) · Zbl 0193.01701  Srivastava H. M., Series Associated with the Zeta and Related Functions (2001) · Zbl 1014.33001  Apostol T. M., Pacific Journal of Mathematics 1 pp 161– (1951)  Srivastava, H. M. Some formulas for the Bernoulli and Euler polynomials at rational arguments. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 129, pp.77–84. · Zbl 0978.11004  DOI: 10.1016/j.jmaa.2005.01.020 · Zbl 1076.33006  DOI: 10.1016/j.camwa.2005.04.018 · Zbl 1099.33011  DOI: 10.1016/S0893-9659(04)90077-8 · Zbl 1070.33012  DOI: 10.1016/0022-247X(88)90326-5 · Zbl 0621.33008  Apostol T. M., Introduction to Analytic Number Theory (1976)  Wang Z. X., Special Functions (1989)  Erdélyi A., Tables of Integral Transforms (1954) · Zbl 0055.36401  Podlubny I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications 198 (1999) · Zbl 0924.34008  Kilbas A. A., Theory and Applications of Fractional Differential Equations 204 (2006) · Zbl 1138.26300  DOI: 10.1016/S0096-3003(03)00746-X · Zbl 1078.11054  Goyal S. P., Ga ita Sandesh 11 pp 99– (1997)  Erdélyi A., Higher Transcendental Functions (1953) · Zbl 0051.30303  Whittaker E. T., A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, 4. ed. (1927) · JFM 53.0180.04
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.