Some properties of the generalized Apostol-type polynomials

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):

(1.1)

(1.2)

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:

(1.6)

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:

(1.8)

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:

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])

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

(1.16)

(1.17)

(1.18)

A polynomial $p_n(x)$ ($n\in \mathbb{N}$, $x\in \mathbb{C}$) is said to be a quasi-monomial [16], whenever two operators $\hat{M}$, $\hat{P}$, called multiplicative and derivative (or lowering) operators respectively, can be defined in such a way that

(1.19)

(1.20)

is analytic function at $t=0$.

From [18], we know that the multiplicative and derivative operators of $R_n(x)$ are

(1.24)

(1.25)

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

Lemma 1.6 ([18])

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])

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.

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

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:

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

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

By using (1.24) and (1.26), we can obtain the multiplicative and derivative operators of the generalized Appostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$

(2.8)

(2.9)

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

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

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:

(2.13)

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:

(2.14)

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.

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.

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

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

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

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.

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.

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

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

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

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.

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.

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

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

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

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.

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.

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

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.

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.

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.

which is equal to (3.8).

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

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

Acting with the operator $expy\frac{{\partial }^2}{\partial x^2}$ on both sides of (3.1), (3.5), (3.13), (3.18), and by using (4.3), we obtain

(4.9)

(4.10)

(4.11)

(4.12)

where ${}_H\mathcal{E}_n^{(\alpha )}(x;\lambda )$ and ${}_H\mathcal{G}_n^{(\alpha )}(x;\lambda )$ are the Hermite-based generalized Apostol-Euler polynomials and the Hermite-based generalized Apostol-Genocchi polynomials respectively, defined by the following generating functions: