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Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials
Advances in Difference Equations volume 2013, Article number: 116 (2013)
Abstract
In this paper, we introduce a unified family of Hermite-based Apostol-Bernoulli, Euler and Genocchi polynomials. We obtain some symmetry identities between these polynomials and the generalized sum of integer powers. We give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials.
MSC:11B68, 33C05.
1 Introduction
Recently, Khan et al. [1] introduced the Hermite-based Appell polynomials via the generating function
where
is the multiplicative operator of the 3-variable Hermite polynomials, which are defined by
and
By using the Berry decoupling identity,
they obtained the generating function of the Hermite-based Appell polynomials as
Letting , they defined Hermite-Bernoulli polynomials by
For , they defined Hermite-Euler polynomials by
and for , they defined Hermite-Genocchi polynomials by
Recently, the author considered the following unification of the Apostol-Bernoulli, Euler and Genocchi polynomials
and obtained the explicit representation of this unified family, in terms of Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given in [2]. Note that the family of polynomials was investigated in [3].
We organize the paper as follows.
In Section 2, we introduce the unification of the Hermite-based generalized Apostol-Bernoulli, Euler and Genocchi polynomials and give summation formulas for this unification. In Section 3, we obtain some symmetry identities for these polynomials. In Section 4, we give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials.
2 Hermite-based generalized Apostol-Bernoulli, Euler and Genocchi polynomials
In this paper, we consider the following general class of polynomials:
For the existence of the expansion, we need
-
(i)
when , and ; when , and ; when , and (or ); , , ; ;
-
(ii)
when ; when ; , , , ; ;
-
(iii)
when and ; , , , ; ,
where , and .
For and in (2.1), we define the following.
Definition 2.1 Let , λ be an arbitrary (real or complex) parameter and . The Hermite-based generalized Apostol-Bernoulli polynomials are defined by
It is clear that
Some special cases of the Hermite-based generalized Apostol-Bernoulli polynomials (some of which are definition) are listed below:
-
is called Hermite-based Apostol-Bernoulli polynomials.
-
is the Hermite-Bernoulli polynomials.
-
is the Apostol-Bernoulli polynomials (see [4–7]). When , we have the classical Bernoulli polynomials.
-
are the Apostol-Bernoulli numbers. gives the classical Bernoulli numbers.
Setting and in (2.1), we get the following.
Definition 2.2 Let α and λ () be an arbitrary (real or complex) parameter and . The Hermite-based generalized Apostol-Euler polynomials are defined by
Obviously, we have
Some special cases of the Hermite-based generalized Apostol-Euler polynomials (some of which are definition) are listed below:
-
is called Hermite-based Apostol-Euler polynomials.
-
is the Hermite-Euler polynomials.
-
is the Apostol-Euler polynomials (see [8]). For , we have the classical Euler polynomials.
-
are the Apostol-Euler numbers. The case gives the classical Euler numbers.
Choosing and in (2.1), we define the following.
Definition 2.3 Let α and λ () be an arbitrary (real or complex) parameter and . The Hermite-based generalized Apostol-Genocchi polynomials are defined by
It is easily seen that
Some special cases of the Hermite-based generalized Apostol-Genocchi polynomials (some of which are definition) are listed below:
-
is called Hermite-based Apostol-Genocchi polynomials.
-
is the Hermite-Genocchi polynomials.
-
is the Apostol-Genocchi polynomials (see [9, 10]). When , we have the classical Genocchi polynomials.
-
are the Apostol-Genocchi numbers. gives the classical Genocchi numbers.
Finally we define the unified Hermite-based Apostol polynomials by
Thus it is clear that and that we have the following observations at once:
-
are the Hermite-based Apostol-Bernoulli polynomials.
-
are the Hermite-based Apostol-Euler polynomials.
-
are the Hermite-based Apostol-Genocchi polynomials.
For the other generalization, we refer [11–25] and [26]. Now we give some relations between the above mentioned Apostol polynomials.
Using (2.1), we get the following identity at once.
Theorem 2.1 Let ; ; be such that the conditions (i)-(iii) are satisfied. Then, the following relation
holds true.
Corollary 2.2 For each , the following relation
holds true for the Hermite-based generalized Apostol-Bernoulli polynomials.
Corollary 2.3 For each , the following relation
holds true for the Hermite-based generalized Apostol-Euler polynomials.
Corollary 2.4 For each , the following relation
holds true for the Hermite-based generalized Apostol-Genocchi polynomials.
Theorem 2.5 For each , the following relation
holds true between the Hermite-based generalized Apostol-Bernoulli and Euler polynomials.
Proof By direct calculations, we have
Comparing the coefficients of on both sides, we get the result. □
3 Symmetry identities for the unified family
For each , the sum is known as the power sum and we have the following generating relation:
For an arbitrary real or complex λ, the generalized sum of integer powers is defined, in [27], via the following generating relation:
It clear that .
For each , the sum is known as the sum of alternative integer powers. The following generating relation is straightforward:
For an arbitrary real or complex λ, the generalized sum of alternative integer powers is defined, in [27], by
Clearly . On the other hand, if n is even, then
We start by obtaining certain symmetry identities, which includes the results given in [28–32] and [27], when .
Theorem 3.1 Let , be such that the conditions (i)-(iii) are satisfied with t replaced by ct and dt. Then we have the following symmetry identity:
Proof Let
Expanding into a series, we get
Now, using Corollary 2 in [[33], p.890], we get
In a similar manner,
From (3.2) and (3.3), we get the result. □
For and we get the following corollary at once.
Corollary 3.2 For all , , , we have the following symmetry identity for the Hermite based generalized Apostol-Bernoulli polynomials:
For and we get, by considering (3.1) that
Corollary 3.3 For all , , , we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d,
Letting and and taking into account (3.1) that we have the following.
Corollary 3.4 For all , , , we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d, that
4 Closed-form formulae for Hermite-based generalized Apostol polynomials
In this section, taking into account the relations
we observe the following fact:
Using (4.1), we start by proving the following closed form summation formula:
Theorem 4.1 Let the conditions (i)-(iii) be satisfied. The following summation formula:
holds true.
Proof Taking logarithms on both sides of (4.1) and then differentiating with respect to t, we get
Inserting the corresponding generating relations, we obtain
and hence
Using the fact that (see [[34], p.101, Lemma 3])
we get
Whence the result. □
Corollary 4.2 Let and . For all , , , we have the following closed form summation formula for the generalized Apostol-Bernoulli polynomials:
Corollary 4.3 Let and . For all , , , we have the following closed form summation formula for the generalized Apostol-Euler polynomials:
Corollary 4.4 Let and . For all , , , we have the following closed form summation formula for the generalized Apostol-Genocchi polynomials:
Theorem 4.5 Let the conditions (i)-(iii) be satisfied. Then we have the following relation between Hermite based Apostol polynomials and 3d-Hermite polynomials:
Proof From (2.1), we can write that
Therefore, we get
Multiplying both sides by , we have
Taking into account (1.1) and (4.3), then using (4.2), we get
Whence the result. □
Corollary 4.6 Let and . For all , , , we have the following summation formula between the Hermite-based generalized Apostol-Bernoulli polynomials and 3d-Hermite polynomials:
Corollary 4.7 Let and . For all , , , we have the following summation formula between the Hermite-based generalized Apostol-Euler polynomials and 3d-Hermite polynomials:
Corollary 4.8 Let and . For all , , , we have the following summation formula between the Hermite-based generalized Apostol-Genocchi polynomials and 3d-Hermite polynomials:
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
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Özarslan, M.A. Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials. Adv Differ Equ 2013, 116 (2013). https://doi.org/10.1186/1687-1847-2013-116
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DOI: https://doi.org/10.1186/1687-1847-2013-116