Abstract
In this paper, we study some properties of the generalized Apostoltype polynomials (see (Luo and Srivastava in Appl. Math. Comput. 217:57025728, 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 ApostolEuler polynomials, the generalized ApostolBernoulli polynomials, and ApostolGenocchi polynomials of high order.
MSC: 11B68, 33C65.
Keywords:
generalized Apostol type polynomials; recurrence relations; differential equations; connected problems; quasimonomial1 Introduction, definitions and motivation
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.532533 and [2], p.61 et seq.; see also [3] and the references cited therein):
and
So that, obviously, the classical Bernoulli polynomials , the classical Euler polynomials and the classical Genocchi polynomials are given, respectively, by
For the classical Bernoulli numbers , the classical Euler numbers and the classical Genocchi numbers of order 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.8384). We begin by recalling here Apostol’s definitions as follows.
Definition 1.1 (Apostol [4]; see also Srivastava [5])
The ApostolBernoulli polynomials () are defined by means of the following generating function:
with, of course,
where denotes the socalled ApostolBernoulli numbers.
Recently, Luo and Srivastava [6] further extended the ApostolBernoulli polynomials as the socalled ApostolBernoulli polynomials of order α.
Definition 1.2 (Luo and Srivastava [6])
The ApostolBernoulli polynomials () of order are defined by means of the following generating function:
with, of course,
where denotes the socalled ApostolBernoulli numbers of order α.
On the other hand, Luo [7], gave an analogous extension of the generalized Euler polynomials as the socalled ApostolEuler polynomials of order α.
Definition 1.3 (Luo [7])
The ApostolEuler polynomials () of order are defined by means of the following generating function:
with, of course,
where denotes the socalled ApostolEuler 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, [814]). Moreover, Luo (see [1214]) introduced and investigated the ApostolGenocchi polynomials of (real or complex) order α, which are defined as follows:
Definition 1.4 The ApostolGenocchi polynomials () of order are defined by means of the following generating function:
with, of course,
where , and denote the socalled ApostolGenocchi numbers, the ApostolGenocchi numbers of order α and the ApostolGenocchi polynomials, respectively.
Recently, Luo and Srivastava [15] introduced a unification (and generalization) of the abovementioned 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:
where
denote the socalled 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 quasimonomial [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
The Appell polynomials [17] can be defined by considering the following generating function:
where
From [18], we know that the multiplicative and derivative operators of are
where
By using (1.21), we have the following lemma.
Lemma 1.6 ([18])
The Appell polynomialsdefined by (1.22) satisfy the differential equation:
where the numerical coefficients, are defined in (1.26), and are linked to the valuesby the following relations:
Let be the vector space of polynomials with coefficients in ℂ. A polynomial sequence be a polynomial set. is called a σAppell polynomial set of transfer power series A is generated by
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 σAppell polynomial sets.
Lemma 1.7 ([19])
Let. Letandbe twoσAppell polynomial sets of transfer power series, respectively, and. Then
where
In recent years, several authors obtained many interesting results involving the related Bernoulli polynomials and Euler polynomials [5,2040]. 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 ApostolEuler polynomials, the generalized ApostolBernoulli polynomials and ApostolGenocchi polynomials of high order.
2 Recursion formulas and differential equations
From the generating function (1.14), we have
A recurrence relation for the generalized Apostol type polynomials is given by the following theorem.
Theorem 2.1For any integral, and, the following recurrence relation for the generalized Apostol type polynomialsholds 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 , and in Theorem 2.1, and then multiplying on both sides of the result, we have:
Corollary 2.2For any integral, and, the following recurrence relation for the generalized ApostolBernoulli polynomialsholds true:
By setting and in Theorem 2.1, we have the following corollary.
Corollary 2.3For any integral, and, the following recurrence relation for the generalized ApostolEuler polynomialsholds true:
By setting and in Theorem 2.1, we have the following corollary.
Corollary 2.4For any integral, and, the following recurrence relation for the generalized ApostolGenocchi polynomialsholds true:
From (1.14) and (1.22), we know that the generalized Appostol type polynomials is Appell polynomials with
From the Eq. (23) of [15], we know that . So from (2.6) and (1.12), we can obtain that if , we have
By using (1.24) and (1.26), we can obtain the multiplicative and derivative operators of the generalized Appostol type polynomials
From (2.1), we can obtain
Then by using (1.20), (2.8) and (2.10), we obtain the following result.
Theorem 2.5For any integral, and, the following recurrence relation for the generalized Apostol type polynomialsholds true:
By setting in Theorem 2.5, we have the following corollary.
Corollary 2.6For any integral, and, the following recurrence relation for the generalized ApostolEuler polynomialsholds true:
Furthermore, applying Lemma 1.7 to , we have the following theorem.
Theorem 2.7The generalized Apostol type polynomialssatisfy the differential equation:
Specially, by setting in Theorem 2.7, then we have the following corollary.
Corollary 2.8The generalized ApostolEuler polynomialssatisfy the differential equation:
3 Connection problems
From (1.14) and (1.28), we know that the generalized Apostol type polynomials are a Appell polynomial set, where denotes the derivative operator.
From Table 1 in [19], we know that the derivative operators of monomials and the GouldHopper polynomials [30] are all . And their transfer power series are 1 and , respectively.
Applying Lemma 1.7 to and , we have the following theorem.
Theorem 3.1
whereis the socalled Apostol type numbers of orderαdefined by (1.15).
By setting , and in Theorem 3.1, and then multiplying on both sides of the result, we have the following corollary.
Corollary 3.2
which is just Eq. (3.1) of[23].
By setting and in Theorem 3.1, we have the following corollary.
Corollary 3.3
By setting and in Theorem 3.1, we have the following corollary.
Corollary 3.4
which is just Eq. (24) of[15].
Applying Lemma 1.7 to and , we have the following theorem.
Theorem 3.5
whereis the socalled Apostol type numbers of orderαdefined by (1.15).
By setting , and in Theorem 3.5, and then multiplying on both sides of the result, we have the following corollary.
Corollary 3.6
which is just Eq. (3.2) of[23].
By setting and in Theorem 3.5, we have the following corollary.
Corollary 3.7
By setting and in Theorem 3.5, we have the following corollary.
Corollary 3.8
Applying Lemma 1.7 to and , we have the following theorem.
Theorem 3.9
By setting , and in Theorem 3.9, and then multiplying on both sides of the result, we have the following corollary.
Corollary 3.10
which is just Eq. (3.3) of[23].
By setting and in Theorem 3.9, we have the following corollary.
Corollary 3.11
By setting and in Theorem 3.9, we have the following corollary.
Corollary 3.12
When , applying Lemma 1.7 to and , we have the following theorem.
By setting , and in Theorem 3.13, and then multiplying on both sides of the result, we have the following corollary.
Corollary 3.14
By setting and in Theorem 3.13, we have the following corollary.
Corollary 3.15
which is just the case ofin (3.4).
When or , applying Lemma 1.7 to and , we can obtain the following theorem.
By setting , and in Theorem 3.13, and then multiplying on both sides of the result, we have the following corollary.
Corollary 3.17
When in (3.17), it is just (3.15).
By setting and in Theorem 3.16, we have the following corollary.
Corollary 3.18
which is equal to (3.8).
Corollary 3.19
4 Hermitebased generalized Apostol type polynomials
Finally, we give a generation of the generalized Apostol type polynomials.
The twovariable HermiteKampé de Fériet polynomials (2VHKdFP) are defined by the series [31]
with the following generating function:
And the 2VHKdFP are also defined through the operational identity
Acting the operator on (1.14), and by the identity [32]
we define the Hermitebased generalized Apostol type polynomials by the generating function
Clearly, we have
From the generating function (4.5), we easily obtain
and
which can be combined to get the identity
Acting with the operator on both sides of (3.1), (3.5), (3.13), (3.18), and by using (4.3), we obtain
where and are the Hermitebased generalized ApostolEuler polynomials and the Hermitebased generalized ApostolGenocchi polynomials respectively, defined by the following generating functions:
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this paper, and read and approved the final manuscript.
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.
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