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Some results for Apostol-type polynomials associated with umbral algebra
Advances in Difference Equations volume 2013, Article number: 201 (2013)
Abstract
A family of the Apostol-type polynomials was introduced and investigated recently by Luo and Srivastava (see (Appl. Math. Comput. 217:5702-5728, 2011)). In this paper, we study this polynomial family on P, the algebra of polynomials in a single variable x over all linear functional on P. By using the way of the umbral algebra, we obtain some fundamental properties of the generalized Apostol-type polynomials. We also show some special cases which include the corresponding results of Dere and Simsek etc.
MSC:05A40, 11B68, 05A10, 05A15.
1 Introduction, definitions and motivation
Throughout this paper, we make use of the following conventional notations: denotes the set of natural numbers, ℂ denotes the set of complex numbers.
The classical Bernoulli polynomials , the classical Euler polynomials and the classical Genocchi polynomials , together with their familiar generalizations , and of order α, are usually defined by means of the following generating functions (see, for details, [[1], pp.532-533] and [2]):
and
It is easy to see that , and 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 [[3], p.165, Eq. (3.1)]) and (more recently) by Srivastava (see [[4], pp.83-84]). We begin by recalling Apostol’s definitions as follows.
Definition 1.1 (Apostol [3]; see also Srivastava [4])
The Apostol-Bernoulli polynomials () are defined by means of the following generating function:
with, of course,
where denotes the so-called Apostol-Bernoulli numbers.
Recently, Luo and Srivastava [5] further extended the Apostol-Bernoulli polynomials as the so-called Apostol-Bernoulli polynomials of order α.
Definition 1.2 (Luo and Srivastava [5])
The Apostol-Bernoulli polynomials () of order α () are defined by means of the following generating function:
with, of course,
where denotes the so-called Apostol-Bernoulli numbers of order α.
In this sequel, Luo [6] gave an analogous extension of the generalized Euler polynomials which is the so-called Apostol-Euler polynomials of order α.
Definition 1.3 (Luo [6])
The Apostol-Euler polynomials of order α () are defined by means of the following generating function:
with, of course,
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, [7–11]). Moreover, Luo (see [12]) 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 α () are defined by means of the following generating function:
with, of course,
where , and denote the so-called Apostol-Genocchi numbers, the Apostol-Genocchi numbers of order α and the Apostol-Genocchi polynomials, respectively.
Ozden et al. [13] introduced and investigated the following unification (and generalization) of the generating functions of the three families of Apostol-type polynomials:
It is found from [14] that Ozden further gave an extension of the above definition (1.14) as follows:
Definition 1.5
The author [15] obtained a unified relation between the and the Gauss hypergeometric function , and gave some identities of .
Recently, Luo and Srivastava [16] introduced more general unification (and generalization) of the above-mentioned three families of the generalized Apostol-type polynomials.
Definition 1.6 (Luo and Srivastava [16])
The generalized Apostol-type polynomials (; ) of order α are defined by means of the following generating function:
Clearly, we have
and
In [5, 6, 17, 18], the authors have researched some elementary properties of the Apostol-type polynomials, and some relationships among the Apostol-type polynomials. More investigations about this subject can be found in [13, 15, 16, 19–30].
The aim of this paper is to study the generalized Apostol-type polynomials on the umbral algebra by using the way as the reference [31–33]. We research some fundamental properties of this polynomial family. Some special cases, which include the corresponding results [31–33], are also considered.
2 Umbral algebra of Roman
We can use the following notations and definitions, which are given by Roman [34], pp.1-125].
Let P be the algebra of polynomials in a single variable x over the field of complex numbers. Let be the vector space of all linear functionals on P. Let be the action of a linear functional L on a polynomial . Let ℱ denote the algebra of formal power series
Such algebra is called umbral algebra. Each defines a linear functional on P and
for all .
The order of a power series is the smallest integer k for which the coefficient of does not vanish. A series for which will be called a delta series. When we are considering a delta series in ℱ as a linear functional, we will refer to it as a delta functional.
It is well known that , where denotes the Kronecker symbol. For all in ℱ,
Let and be in ℱ. Then we have
For , then the evaluation functional is defined to be the power series . By (2.2), we have
for all in P. The forward difference functional is the delta functional and
The Abel functional is the delta functional . We have
The Sheffer polynomials are defined by means of the following generating function
Roman [34] proved the following theorem which is represented by the Sheffer polynomials (or Sheffer sequences) explicitly.
Theorem 2.1 Let be a delta series and let be an invertible series. Then there exists a unique sequence of polynomials satisfying the orthogonality conditions
for all .
The sequence in (2.7) is the Sheffer polynomials for pair , where must be invertible and must be delta series. The Sheffer polynomials for pair is the Appell polynomials or the Appell sequences for .
The Appell polynomials, the Bernoulli polynomials, the Euler polynomials, the Genocchi polynomials and the Genocchi polynomials of higher order belong to the family of the Sheffer polynomials (cf. [31, 34–36]).
The Sheffer polynomials satisfy the following relations:
derivative formula
recurrence formula
expansion theorem
multiplication theorem, for ,
and
3 The Apostol-type polynomials on ℱ
We see from Definition 1.6 and (2.6) that the generalized Apostol-type polynomials also belong to the Sheffer polynomials where .
In this section, by using the properties of the Sheffer sequences and also the Appell sequences, we prove many fundamental properties of the generalized Apostol-type polynomials defined by (1.16).
By using (2.8) and (1.16), we arrive at the following lemma.
Lemma 3.1
Theorem 3.2
where and denote the first-order generalized Apostol-type polynomials and the Stirling numbers of the second kind, respectively.
Proof By Lemma 3.1, we obtain
By using (2.3) and (2.9), we get
Setting
where denotes the Stirling numbers of second kind (cf. [[34], p.59]) in (3.3), we arrive at the desired result. □
We deduce the following formulas.
Letting , taking and in (3.2) and noting relation (1.17), we deduce the following result.
Corollary 3.3 (see [[32], Remark 19])
where and denote the Apostol-Bernoulli polynomials and the Stirling numbers of the second kind, respectively.
Taking and in (3.2) and noting relation (1.18), we deduce the following result.
Corollary 3.4 (see [[32], Remark 21])
where and denote the Apostol-Euler polynomials and the Stirling numbers of the second kind, respectively.
Taking in (3.2) and noting relation (1.19), we deduce the following result.
Corollary 3.5 (see [[32], Remark 20])
where and denote the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, respectively.
Setting in (3.6), we deduce Theorem 2 in the work [[31], p.758, Theorem 2].
Corollary 3.6
where and denote the Genocchi polynomials and the Stirling numbers of the second kind, respectively.
Letting , taking , , in (3.2) and noting relation (1.20), thus we deduce the following formulas of the polynomials .
Corollary 3.7
where and denote the generalization of Apostol type polynomials defined by (1.14) and the Stirling numbers of the second kind, respectively.
By using (2.9), we arrive at the following lemma.
Lemma 3.8
Remark 3.9 An alternative proof of Lemma 3.8 is also obtained from (1.16) by using derivative with respect to x. By Lemma 3.8, one can see that
Theorem 3.10
Proof By Lemma 3.1, we obtain
After some calculations in the above equation, we have
Using (1.16) and (3.10), we obtain the desired result. □
Letting , taking and in (3.11) and noting relation (1.17), we deduce the following result.
Corollary 3.11 (see [[32], Remark 32])
Taking and in (3.11) and noting relation (1.18), we deduce the following result.
Corollary 3.12 (see [[32], Remark 33])
Taking in (3.11) and noting relation (1.19), we deduce the following result.
Corollary 3.13 (see [[32], Remark 34])
Setting in the above equation, we deduce Lemma 3 in [[31], p.758].
Corollary 3.14
Taking , , in (3.11) and noting relation (1.21), we deduce
Corollary 3.15
An integral representation of is given by the following theorem.
Theorem 3.16
Proof By using Lemma 3.8, we have
By (2.3), we obtain
Using (2.5), we obtain the desired result. □
Setting in (3.19) and noting relation (1.19), we deduce the Theorem 3 in [[31], p.758].
Corollary 3.17
A recurrence formula for is given by the next theorem.
Theorem 3.18 (Recurrence formula)
Proof Setting
in (2.10), one can obtain
By using Theorem 3.10 and (3.10), we have
After some calculations in the above equation, we get the desired result. □
Letting , taking and in (3.21) and noting relation (1.17), we deduce the following known result.
Corollary 3.19 (see, e.g., [[32], Remark 38])
Taking and in (3.21) and noting relation (1.18), we deduce the following known result.
Corollary 3.20 (see, e.g., [[32], Remark 39])
Taking in (3.21) and noting relation (1.19), we deduce the following known result.
Corollary 3.21 (see, e.g., [[32], Remark 40])
Setting in the above equation, we have the following.
Corollary 3.22 (see [[31], p.759, Theorem 4])
Taking , , in (3.21) and noting relation (1.21), thus we deduce the following result.
Corollary 3.23
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Acknowledgements
Dedicated to Professor Hari M. Srivastava on the occasion of his seventy-third birth anniversary. The authors sincerely thank the referees for their valuable suggestions and comments. The present investigation was supported by the National Natural Science Foundation of China under Grant 11226281, Fund of Science and Innovation of Yangzhou University, China under Grant 2012CXJ005, Natural Science Foundation Project of Chongqing, China under Grant CSTC2011JJA00024, Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625, Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07 and Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308.
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Lu, DQ., Xiang, CH. & Luo, QM. Some results for Apostol-type polynomials associated with umbral algebra. Adv Differ Equ 2013, 201 (2013). https://doi.org/10.1186/1687-1847-2013-201
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DOI: https://doi.org/10.1186/1687-1847-2013-201