The classical Bernoulli polynomials , the classical Euler polynomials and the classical Genocchi polynomials , together with their familiar generalizations , and 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
(1.3)
So that, obviously, the classical Bernoulli polynomials , the classical Euler polynomials and the classical Genocchi polynomials are given, respectively, by
(1.4)
For the classical Bernoulli numbers , the classical Euler numbers and the classical Genocchi numbers of order n, we have
(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 () are defined by means of the following generating function:
with, of course,
(1.7)
where 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 () of order are defined by means of the following generating function:
with, of course,
(1.9)
where 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 () of order are defined by means of the following generating function:
(1.10)
with, of course,
(1.11)
where denotes the so-called Apostol-Euler numbers of order α.
On the subject of the Genocchi polynomials 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 () of order are defined by means of the following generating function:
(1.12)
with, of course,
(1.13)
where , and 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 (, ) of order α are defined by means of the following generating function:
(1.14)
where
(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 (, ) is said to be a quasi-monomial [16], whenever two operators , , called multiplicative and derivative (or lowering) operators respectively, can be defined in such a way that
which can be combined to get the identity
(1.21)
The Appell polynomials [17] can be defined by considering the following generating function:
(1.22)
where
(1.23)
is analytic function at .
From [18], we know that the multiplicative and derivative operators of are
where
(1.26)
By using (1.21), we have the following lemma.
Lemma 1.6 ([18])
The Appell polynomials defined by (1.22) satisfy the differential equation:
(1.27)
where the numerical coefficients , are defined in (1.26), and are linked to the values by the following relations:
Let be the vector space of polynomials with coefficients in ℂ. A polynomial sequence be a polynomial set. is called a σ-Appell polynomial set of transfer power series A is generated by
(1.28)
where 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 . Let and be two σ-Appell polynomial sets of transfer power series, respectively, and . Then
(1.29)
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 , 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.