The classical Bernoulli polynomials

${B}_{n}(x)$, the classical Euler polynomials

${E}_{n}(x)$ and the classical Genocchi polynomials

${G}_{n}(x)$, together with their familiar generalizations

${B}_{n}^{(\alpha )}(x)$,

${E}_{n}^{(\alpha )}(x)$ and

${G}_{n}^{(\alpha )}(x)$ of (real or complex) order

*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):

and

${\left(\frac{2z}{{e}^{z}+1}\right)}^{\alpha}\cdot {e}^{xz}=\sum _{n=0}^{\mathrm{\infty}}{G}_{n}^{(\alpha )}(x)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}(|z|<\pi ).$

(1.3)

So that, obviously, the classical Bernoulli polynomials

${B}_{n}(x)$, the classical Euler polynomials

${E}_{n}(x)$ and the classical Genocchi polynomials

${G}_{n}(x)$ are given, respectively, by

${B}_{n}(x):={B}_{n}^{(1)}(x),\phantom{\rule{2em}{0ex}}{E}_{n}(x):={E}_{n}^{(1)}(x)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{G}_{n}(x):={G}_{n}^{(1)}(x)\phantom{\rule{1em}{0ex}}(n\in {\mathbb{N}}_{0}).$

(1.4)

For the classical Bernoulli numbers

${B}_{n}$, the classical Euler numbers

${E}_{n}$ and the classical Genocchi numbers

${G}_{n}$ of order

*n*, we have

${B}_{n}:={B}_{n}(0)={B}_{n}^{(1)}(0),\phantom{\rule{2em}{0ex}}{E}_{n}:={E}_{n}(0)={E}_{n}^{(1)}(0)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{G}_{n}:={G}_{n}(0)={G}_{n}^{(1)}(0),$

(1.5)

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])

The Apostol-Bernoulli polynomials

${\mathcal{B}}_{n}(x;\lambda )$ (

$\lambda \in \mathbb{C}$) are defined by means of the following generating function:

with, of course,

${B}_{n}(x)={\mathcal{B}}_{n}(x;1)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{B}}_{n}(\lambda ):={\mathcal{B}}_{n}(0;\lambda ),$

(1.7)

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])

The Apostol-Bernoulli polynomials

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

$\lambda \in \mathbb{C}$) of order

$\alpha \in {\mathbb{N}}_{0}$ are defined by means of the following generating function:

with, of course,

${B}_{n}^{(\alpha )}(x)={\mathcal{B}}_{n}^{(\alpha )}(x;1)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{B}}_{n}^{(\alpha )}(\lambda ):={\mathcal{B}}_{n}^{(\alpha )}(0;\lambda ),$

(1.9)

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])

The Apostol-Euler polynomials

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

$\lambda \in \mathbb{C}$) of order

$\alpha \in {\mathbb{N}}_{0}$ are defined by means of the following generating function:

${\left(\frac{2}{\lambda {e}^{z}+1}\right)}^{\alpha}\cdot {e}^{xz}=\sum _{n=0}^{\mathrm{\infty}}{\mathcal{E}}_{n}^{(\alpha )}(x;\lambda )\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}(|z|<|log(-\lambda )|)$

(1.10)

with, of course,

${E}_{n}^{(\alpha )}(x)={\mathcal{E}}_{n}^{(\alpha )}(x;1)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{E}}_{n}^{(\alpha )}(\lambda ):={\mathcal{E}}_{n}^{(\alpha )}(0;\lambda ),$

(1.11)

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:

${\left(\frac{2z}{\lambda {e}^{z}+1}\right)}^{\alpha}\cdot {e}^{xz}=\sum _{n=0}^{\mathrm{\infty}}{\mathcal{G}}_{n}^{(\alpha )}(x;\lambda )\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}(|z|<|log(-\lambda )|)$

(1.12)

with, of course,

$\begin{array}{r}{G}_{n}^{(\alpha )}(x)={\mathcal{G}}_{n}^{(\alpha )}(x;1),\phantom{\rule{2em}{0ex}}{\mathcal{G}}_{n}^{(\alpha )}(\lambda ):={\mathcal{G}}_{n}^{(\alpha )}(0;\lambda ),\\ {\mathcal{G}}_{n}(x;\lambda ):={\mathcal{G}}_{n}^{(1)}(x;\lambda )\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{G}}_{n}(\lambda ):={\mathcal{G}}_{n}^{(1)}(\lambda ),\end{array}$

(1.13)

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])

The generalized Apostol type polynomials

${\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)$ (

$\alpha \in {\mathbb{N}}_{0}$,

$\lambda ,u,v\in \mathbb{C}$) of order

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

${\left(\frac{{2}^{u}{z}^{v}}{\lambda {e}^{z}+1}\right)}^{\alpha}\cdot {e}^{xz}=\sum _{n=0}^{\mathrm{\infty}}{\mathcal{F}}_{n}^{(\alpha )}(x;\lambda ;u,v)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}(|z|<|log(-\lambda )|),$

(1.14)

where

${\mathcal{F}}_{n}^{(\alpha )}(\lambda ;u,v):={\mathcal{F}}_{n}^{(\alpha )}(0;\lambda ;u,v)$

(1.15)

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

So that, by comparing Definition 1.5 with Definitions 1.2, 1.3 and 1.4, we have

A polynomial

${p}_{n}(x)$ (

$n\in \mathbb{N}$,

$x\in \mathbb{C}$) is said to be a quasi-monomial [

16], whenever two operators

$\stackrel{\u02c6}{M}$,

$\stackrel{\u02c6}{P}$, called multiplicative and derivative (or lowering) operators respectively, can be defined in such a way that

which can be combined to get the identity

$\stackrel{\u02c6}{M}\stackrel{\u02c6}{P}{p}_{n}(x)=n{p}_{n}(x).$

(1.21)

The Appell polynomials [

17] can be defined by considering the following generating function:

$A(t){e}^{xt}=\sum _{n=0}^{\mathrm{\infty}}\frac{{R}_{n}(x)}{n!}{t}^{n},$

(1.22)

where

$A(t)=\sum _{k=0}^{\mathrm{\infty}}\frac{{R}_{k}}{k!}{t}^{k}\phantom{\rule{1em}{0ex}}(A(0)\ne 0)$

(1.23)

is analytic function at $t=0$.

From [

18], we know that the multiplicative and derivative operators of

${R}_{n}(x)$ are

where

$\frac{{A}^{\prime}(t)}{A(t)}=\sum _{n=0}^{\mathrm{\infty}}{\alpha}_{n}\frac{{t}^{n}}{n!}.$

(1.26)

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*:

$\frac{{\alpha}_{n-1}}{(n-1)!}{y}^{(n)}+\frac{{\alpha}_{n-2}}{(n-2)!}{y}^{(n-1)}+\cdots +\frac{{\alpha}_{1}}{1!}{y}^{\u2033}+(x+{\alpha}_{0}){y}^{\prime}-ny=0,$

(1.27)

*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*:

${R}_{k+1}=\sum _{h=0}^{k}\left(\begin{array}{c}k\\ h\end{array}\right){R}_{h}{\alpha}_{k-h}.$

Let

$\mathcal{P}$ be the vector space of polynomials with coefficients in ℂ. A polynomial sequence

${\{{P}_{n}\}}_{n\ge 0}$ be a polynomial set.

${\{{P}_{n}\}}_{n\ge 0}$ is called a

*σ*-Appell polynomial set of transfer power series

*A* is generated by

$G(x,t)=A(t){G}_{0}(x,t)=\sum _{n=0}^{\mathrm{\infty}}\frac{{P}_{n}(x)}{n!}{t}^{n},$

(1.28)

where

${G}_{0}(x,t)$ is a solution of the system:

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* ${Q}_{n}(x)=\sum _{m=0}^{n}\frac{n!}{m!}{\alpha}_{n-m}{P}_{m}(x),$

(1.29)

*where*
$\frac{{A}_{2}(t)}{{A}_{1}(t)}=\sum _{k=0}^{\mathrm{\infty}}{\alpha}_{k}{t}^{k}.$

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.