𝑞 -Extensions for the Apostol Type Polynomials

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


Introduction, Preliminaries, and Definitions
Throughout this research we always apply the following notations. N indicates the set of natural numbers, N 0 indicates the set of nonnegative integers, R indicates the set of all real numbers, and C denotes the set of complex numbers. We refer the readers to [1] for all the following -standard notations. The -shifted factorial is defined as ( ; ) 0 = 1, 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:

Journal of Applied Mathematics
As an immediate result of these two definitions, we have ( ) (− ) = 1.
Clearly, for = 1, one has

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.
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.
Proof. We only prove (18). By using Definition 2 and starting from the left hand side of the relation (18) Comparing the coefficients of the term /[ ] ! in both sides gives the result.

-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     Journal of Applied Mathematics Taking = 1 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
Letting ↑ 1, we get the Luo-Srivastava addition theorem (see [12]): Proof. The proof follows from the following identity: (38) Proof. The -Stirling polynomials ( , ) of the second kind are defined by means of the following generating function: