# Some properties of the generalized Apostol-type polynomials

- Da-Qian Lu
^{1}and - Qiu-Ming Luo
^{2}Email author

**2013**:64

**DOI: **10.1186/1687-2770-2013-64

© Lu and Luo; licensee Springer. 2013

**Received: **10 December 2012

**Accepted: **19 March 2013

**Published: **28 March 2013

## Abstract

In this paper, we study some properties of the generalized Apostol-type polynomials (see (Luo and Srivastava in Appl. Math. Comput. 217:5702-5728, 2011)), including the recurrence relations, the differential equations and some other connected problems, which extend some known results. We also deduce some properties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli polynomials, and Apostol-Genocchi polynomials of high order.

**MSC:**11B68, 33C65.

### Keywords

generalized Apostol type polynomials recurrence relations differential equations connected problems quasi-monomial## 1 Introduction, definitions and motivation

*α*, are usually defined by means of the following generating functions (see, for details, [1], pp.532-533 and [2], p.61

*et seq*.; see also [3] and the references cited therein):

*n*, we have

respectively.

Some interesting analogues of the classical Bernoulli polynomials and numbers were first investigated by Apostol (see [4], p.165, Eq. (3.1)) and (more recently) by Srivastava (see [5], pp.83-84). We begin by recalling here Apostol’s definitions as follows.

**Definition 1.1** (Apostol [4]; see also Srivastava [5])

where ${\mathcal{B}}_{n}(\lambda )$ denotes the so-called Apostol-Bernoulli numbers.

Recently, Luo and Srivastava [6] further extended the Apostol-Bernoulli polynomials as the so-called Apostol-Bernoulli polynomials of order *α*.

**Definition 1.2** (Luo and Srivastava [6])

where ${\mathcal{B}}_{n}^{(\alpha )}(\lambda )$ denotes the so-called Apostol-Bernoulli numbers of order *α*.

On the other hand, Luo [7], gave an analogous extension of the generalized Euler polynomials as the so-called Apostol-Euler polynomials of order *α*.

**Definition 1.3** (Luo [7])

where ${\mathcal{E}}_{n}^{(\alpha )}(\lambda )$ denotes the so-called Apostol-Euler numbers of order *α*.

On the subject of the Genocchi polynomials ${G}_{n}(x)$ and their various extensions, a remarkably large number of investigations have appeared in the literature (see, for example, [8–14]). Moreover, Luo (see [12–14]) introduced and investigated the Apostol-Genocchi polynomials of (real or complex) order *α*, which are defined as follows:

**Definition 1.4**The Apostol-Genocchi polynomials ${\mathcal{G}}_{n}^{(\alpha )}(x;\lambda )$ ($\lambda \in \mathbb{C}$) of order $\alpha \in {\mathbb{N}}_{0}$ are defined by means of the following generating function:

where ${\mathcal{G}}_{n}(\lambda )$, ${\mathcal{G}}_{n}^{(\alpha )}(\lambda )$ and ${\mathcal{G}}_{n}(x;\lambda )$ denote the so-called Apostol-Genocchi numbers, the Apostol-Genocchi numbers of order *α* and the Apostol-Genocchi polynomials, respectively.

Recently, Luo and Srivastava [15] introduced a unification (and generalization) of the above-mentioned three families of the generalized Apostol type polynomials.

**Definition 1.5** (Luo and Srivastava [15])

*α*are defined by means of the following generating function:

denote the so-called Apostol type numbers of order *α*.

is analytic function at $t=0$.

By using (1.21), we have the following lemma.

**Lemma 1.6** ([18])

*The Appell polynomials*${R}_{n}(x)$

*defined by*(1.22)

*satisfy the differential equation*:

*where the numerical coefficients*${\alpha}_{k}$, $k=1,2,\dots ,n-1$

*are defined in*(1.26),

*and are linked to the values*${R}_{k}$

*by the following relations*:

*σ*-Appell polynomial set of transfer power series

*A*is generated by

In [19], the authors investigated the connection coefficients between two polynomials. And there is a result about connection coefficients between two *σ*-Appell polynomial sets.

**Lemma 1.7** ([19])

*Let*$\sigma \in {\mathrm{\Lambda}}^{(-1)}$.

*Let*${\{{P}_{n}\}}_{n\ge 0}$

*and*${\{{Q}_{n}\}}_{n\ge 0}$

*be two*

*σ*-

*Appell polynomial sets of transfer power series*,

*respectively*, ${A}_{1}$

*and*${A}_{2}$.

*Then*

*where*

In recent years, several authors obtained many interesting results involving the related Bernoulli polynomials and Euler polynomials [5, 20–40]. And in [29], the authors studied some series identities involving the generalized Apostol type and related polynomials.

In this paper, we study some other properties of the generalized Apostol type polynomials ${\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$, including the recurrence relations, the differential equations and some connection problems, which extend some known results. As special, we obtain some properties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli polynomials and Apostol-Genocchi polynomials of high order.

## 2 Recursion formulas and differential equations

A recurrence relation for the generalized Apostol type polynomials is given by the following theorem.

**Theorem 2.1**

*For any integral*$n\ge 1$, $\lambda \in \mathbb{C}$

*and*$\alpha \in \mathbb{N}$,

*the following recurrence relation for the generalized Apostol type polynomials*${\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$

*holds true*:

*Proof* Differentiating both sides of (1.14) with respect to *t*, and using some elementary algebra and the identity principle of power series, recursion (2.2) easily follows. □

By setting $\lambda :=-\lambda $, $u=0$ and $v=1$ in Theorem 2.1, and then multiplying ${(-1)}^{\alpha}$ on both sides of the result, we have:

**Corollary 2.2**

*For any integral*$n\ge 1$, $\lambda \in \mathbb{C}$

*and*$\alpha \in \mathbb{N}$,

*the following recurrence relation for the generalized Apostol*-

*Bernoulli polynomials*${\mathcal{B}}_{n}^{(\alpha )}(x;\lambda )$

*holds true*:

By setting $u=1$ and $v=0$ in Theorem 2.1, we have the following corollary.

**Corollary 2.3**

*For any integral*$n\ge 1$, $\lambda \in \mathbb{C}$

*and*$\alpha \in \mathbb{N}$,

*the following recurrence relation for the generalized Apostol*-

*Euler polynomials*${\mathcal{E}}_{n}^{(\alpha )}(x;\lambda )$

*holds true*:

By setting $u=1$ and $v=1$ in Theorem 2.1, we have the following corollary.

**Corollary 2.4**

*For any integral*$n\ge 1$, $\lambda \in \mathbb{C}$

*and*$\alpha \in \mathbb{N}$,

*the following recurrence relation for the generalized Apostol*-

*Genocchi polynomials*${\mathcal{G}}_{n}^{(\alpha )}(x;\lambda )$

*holds true*:

Then by using (1.20), (2.8) and (2.10), we obtain the following result.

**Theorem 2.5**

*For any integral*$n\ge 1$, $\lambda \in \mathbb{C}$

*and*$\alpha \in \mathbb{N}$,

*the following recurrence relation for the generalized Apostol type polynomials*${\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,0)$

*holds true*:

By setting $u=1$ in Theorem 2.5, we have the following corollary.

**Corollary 2.6**

*For any integral*$n\ge 1$, $\lambda \in \mathbb{C}$

*and*$\alpha \in \mathbb{N}$,

*the following recurrence relation for the generalized Apostol*-

*Euler polynomials*${\mathcal{E}}_{n}^{(\alpha )}(x;\lambda )$

*holds true*:

Furthermore, applying Lemma 1.7 to ${\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,0)$, we have the following theorem.

**Theorem 2.7**

*The generalized Apostol type polynomials*${\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,0)$

*satisfy the differential equation*:

Specially, by setting $u=1$ in Theorem 2.7, then we have the following corollary.

**Corollary 2.8**

*The generalized Apostol*-

*Euler polynomials*${\mathcal{E}}_{n}^{(\alpha )}(x;\lambda )$

*satisfy the differential equation*:

## 3 Connection problems

From (1.14) and (1.28), we know that the generalized Apostol type polynomials ${\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$ are a ${D}_{x}$-Appell polynomial set, where ${D}_{x}$ denotes the derivative operator.

From Table 1 in [19], we know that the derivative operators of monomials ${x}^{n}$ and the Gould-Hopper polynomials ${g}_{n}^{m}(x,h)$ [30] are all ${D}_{x}$. And their transfer power series $A(t)$ are 1 and ${e}^{h{t}^{m}}$, respectively.

Applying Lemma 1.7 to ${P}_{n}(x)={x}^{n}$ and ${Q}_{n}(x)={\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

**Theorem 3.1**

*where* ${\mathcal{F}}_{n}^{(\alpha )}(\lambda ;u,v)$ *is the so*-*called Apostol type numbers of order* *α* *defined by* (1.15).

By setting $\lambda :=-\lambda $, $u=0$ and $v=1$ in Theorem 3.1, and then multiplying ${(-1)}^{\alpha}$ on both sides of the result, we have the following corollary.

**Corollary 3.2**

*which is just Eq*. (3.1) *of* [23].

By setting $u=0$ and $v=0$ in Theorem 3.1, we have the following corollary.

**Corollary 3.3**

By setting $u=1$ and $v=1$ in Theorem 3.1, we have the following corollary.

**Corollary 3.4**

*which is just Eq*. (24) *of* [15].

Applying Lemma 1.7 to ${P}_{n}(x)={\mathcal{F}}_{n}(x;\lambda ;u,v)$ and ${Q}_{n}(x)={\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

**Theorem 3.5**

*where* ${\mathcal{F}}_{n}^{(\alpha )}(\lambda ;u,v)$ *is the so*-*called Apostol type numbers of order* *α* *defined by* (1.15).

By setting $\lambda :=-\lambda $, $u=0$ and $v=1$ in Theorem 3.5, and then multiplying ${(-1)}^{\alpha}$ on both sides of the result, we have the following corollary.

**Corollary 3.6**

*which is just Eq*. (3.2) *of* [23].

By setting $u=1$ and $v=0$ in Theorem 3.5, we have the following corollary.

**Corollary 3.7**

By setting $u=1$ and $v=1$ in Theorem 3.5, we have the following corollary.

**Corollary 3.8**

Applying Lemma 1.7 to ${P}_{n}(x)={g}_{n}^{m}(x,h)$ and ${Q}_{n}(x)={\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

**Theorem 3.9**

By setting $\lambda :=-\lambda $, $u=0$ and $v=1$ in Theorem 3.9, and then multiplying ${(-1)}^{\alpha}$ on both sides of the result, we have the following corollary.

**Corollary 3.10**

*which is just Eq*. (3.3) *of* [23].

By setting $u=1$ and $v=0$ in Theorem 3.9, we have the following corollary.

**Corollary 3.11**

By setting $u=1$ and $v=1$ in Theorem 3.9, we have the following corollary.

**Corollary 3.12**

When $v\alpha =1$, applying Lemma 1.7 to ${P}_{n}(x)={\mathcal{E}}_{n}^{(\alpha -1)}(x;\lambda )$ and ${Q}_{n}(x)={\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

**Theorem 3.13**

*If*$v\alpha =1$,

*then we have*

By setting $\lambda :=-\lambda $, $u=0$ and $v=1$ in Theorem 3.13, and then multiplying ${(-1)}^{\alpha}$ on both sides of the result, we have the following corollary.

**Corollary 3.14**

By setting $u=1$ and $v=1$ in Theorem 3.13, we have the following corollary.

**Corollary 3.15**

*which is just the case of* $\alpha =1$ *in* (3.4).

When $v=1$ or $\alpha =0$, applying Lemma 1.7 to ${P}_{n}(x)={\mathcal{G}}_{n}^{(\alpha -1)}(x;\lambda )$ and ${Q}_{n}(x)={\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$, we can obtain the following theorem.

**Theorem 3.16**

*If*$v=1$

*or*$\alpha =0$,

*we have*

By setting $\lambda :=-\lambda $, $u=0$ and $v=1$ in Theorem 3.13, and then multiplying ${(-1)}^{\alpha}$ on both sides of the result, we have the following corollary.

**Corollary 3.17**

When $\alpha =1$ in (3.17), it is just (3.15).

By setting $u=1$ and $v=1$ in Theorem 3.16, we have the following corollary.

**Corollary 3.18**

*which is equal to* (3.8).

If $\alpha =0$ in Theorem 3.16, we have:

**Corollary 3.19**

## 4 Hermite-based generalized Apostol type polynomials

Finally, we give a generation of the generalized Apostol type polynomials.

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The present investigation was supported, in part, by the *National Natural Science Foundation of China* under Grant 11226281, *Fund of Science and Innovation of Yangzhou University, China* under Grant 2012CXJ005, *Research Project of Science and Technology of Chongqing Education Commission, China* under Grant KJ120625 and *Fund of Chongqing Normal University, China* under Grant 10XLR017 and 2011XLZ07.

## Authors’ Affiliations

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