Research Article | Open Access

# -Extensions for the Apostol Type Polynomials

**Academic Editor:**Necdet Batir

#### Abstract

The aim of this work is to introduce an extension for -standard notations. The -Apostol type polynomials and study some of their properties. Besides, some relations between the mentioned polynomials and some other known polynomials are obtained.

#### 1. Introduction, Preliminaries, and Definitions

Throughout this research we always apply the following notations. indicates the set of natural numbers, indicates the set of nonnegative integers, indicates the set of all real numbers, and denotes the set of complex numbers. We refer the readers to [1] for all the following -standard notations. The -shifted factorial is defined as The -numbers and -factorials are defined by respectively. The -polynomial coefficient is defined by The -analogue of the function is defined by The -binomial formula is known as In the standard approach to the -calculus, two exponential functions are used: As an immediate result of these two definitions, we have .

Recently, Luo and Srivastava [2] introduced and studied the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Euler polynomials . Kurt [3] gave the generalization of the Bernoulli polynomials of order and studied their properties. They also studied these polynomials systematically; see [2, 4–9]. There are numerous recent investigations on this subject by many other authors; see [3, 10–20]. More recently, Tremblay et al. [10] further gave the definition of and studied their properties. On the other hand, Mahmudov and Keleshteri [21, 22] studied various two dimensional -polynomials. Motivated by these papers, we define generalized Apostol type -polynomials as follows.

*Definition 1. *Let , , and. The generalized -Apostol-Bernoulli numbers and polynomials in of order are defined, in a suitable neighborhood of , by means of the generating functions:
where .

*Definition 2. *Let , , and . The generalized -Apostol-Euler numbers and polynomials in of order are defined, in a suitable neighborhood of , by means of the generating functions:

*Definition 3. *Let , and . The generalized -Apostol-Genocchi numbers and polynomials in of order are defined, in a suitable neighborhood of , by means of the generating functions:

Clearly, for , one has For and , one has For , one has

#### 2. Properties of the Apostol Type -Polynomials

In this section, we show some basic properties of the generalized -polynomials. We only prove the facts for one of them. Obviously, by applying the similar technique, other ones can be proved.

Proposition 4. *The generalized -polynomials , , and satisfy the following relations:
*

*Proof. *We only prove the second identity. By using Definition 2, we have
Comparing the coefficients of the term in both sides gives the result.

Corollary 5. *The generalized -polynomials , , and satisfy the following relations:
*

Proposition 6. *The generalized -polynomials , , and satisfy the following relations:
*

*Proof. *We only prove (18). By using Definition 2 and starting from the left hand side of the relation (18), we have
Comparing the coefficients of the term in both sides gives the result.

#### 3. -Analogue of the Luo-Srivastava Addition Theorem

In this section, we state and prove a -generalization of the Luo-Srivastava addition theorem.

Theorem 7. *The following relation holds between generalized -Apostol-Euler and -Apostol-Bernoulli polynomials:
*

*Proof. *We take aid of the following identity to prove (21):
Therefore, we can write
From that we can conclude the following:
That is,
Substituting (25) into the right hand side of (16), we obtain
Thus, from one hand, we can write
As we know that
we can continue as
On the other hand, for , we can write
and, as , we have
Adding to we obtain
Consequently,

Taking in Theorem 7, we get a -generalization of the Luo-Srivastava addition theorem [2].

Corollary 8. *The following relation holds between generalized -Apostol-Euler and -Apostol-Bernoulli polynomials:
*

Letting , we get the Luo-Srivastava addition theorem (see [12]):

Next theorem gives relationship between and .

Theorem 9. *The following relation holds between generalized -Apostol-Euler and -Apostol-Genocchi polynomials:
*

*Proof. *The proof follows from the following identity:

Theorem 10. *The following relation holds between generalized -Apostol-Euler and -Stirling polynomials of the second kind:
*

*Proof. *The -Stirling polynomials of the second kind are defined by means of the following generating function:
where ; see [23]. Replacing identity (39) in the right hand side of (16), we have

Theorem 11. *The relationship
**
holds between the polynomials and the -Hermite polynomials defined by (see [24])
*

*Proof. *Indeed,

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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#### Copyright

Copyright © 2014 Nazim I. Mahmudov and Marzieh Eini Keleshteri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.