Abstract

Through a modification on the parameters associated with generating function of the -extensions for the Apostol type polynomials of order and level , we obtain some new results related to a unified presentation of the -analog of the generalized Apostol type polynomials of order and level . In addition, we introduce some algebraic and differential properties for the -analog of the generalized Apostol type polynomials of order and level and the relation of these with the -Stirling numbers of the second kind, the generalized -Bernoulli polynomials of level , the generalized -Apostol type Bernoulli polynomials, the generalized -Apostol type Euler polynomials, the generalized -Apostol type Genocchi polynomials of order and level , and the -Bernstein polynomials.

1. Introduction

With the development of -calculus in the mid-19th century, many authors made generalizations to special functions and polynomial families based on the -analogs (cf. [1–7]). During this process, properties and relations have been demonstrated and contributed to solving different kinds of problems in other subjects (see, [8–10]).

In 2003, Natalini P. and Bernardini A. [11] introduced a class the polynomials , considering a class of Appell polynomials defined by using a generating function linked to the Mittag-Leffler function (see, [12, p. 204, Eq. ])Kurt B. [13] did the generalization of the Bernoulli , Euler , and Genocchi polynomials of order and level . Tremblay R. et al. [14, 15] defined the generalized Apostol-Bernoulli polynomials and their properties. In [12] which has studied the unification of Bernoulli, Euler, and Genocchi polynomials they considered the following Mittag-Leffler type function (see, [12, p. 209, Eq. ]):to define a extension of the generalized Apostol type polynomials in , parameters , order , and level through the following generating function:where , , and The numbers were given by Recently a new class of polynomials has been introduced in [16] and it provides a unification of three families of polynomials through the following generating function (see, [16, p. 923, Eq. ]): where , , and . The author named it the unified -Apostol-Bernoulli, Euler, and Genocchi polynomials of order and proved some properties for these unification.

On the other hand, in the definition of the -Mittag-Leffler function when , , and lead us to (see, [17, p. 614, Eq ])which corresponds to the -analog of the generating function defined in (1) and with this, new research emerged about other polynomial families based on the -analogs.

In [18], the authors introduced the generalized -Apostol-Bernoulli polynomials, the generalized -Apostol Euler polynomials, and generalized -Apostol Genocchi polynomials in variable , order , and level through the following generating functions, defined in a suitable neighborhood of (see, [18, p. 2 Eq , , ])where , , , and .

Based on the previous result, we focus our attention on a new unification of the -analog of the generalized Apostol type polynomials of order and level , defined in [18], by doing some modifications to the generating function linked to the -Mittag-Leffler function (6) through new parameters following the same scheme or procedure applied by [12].

The paper is organized as follows. Section 2 contains some notations, definitions, and properties of the -analogs and some results about -analogs of the Apostol type polynomials. In Section 3, we introduce the unification -analog of the generalized Apostol type polynomials in , parameters , order , and level and their algebraic and differential properties. Finally in the Section 4, we show relations between the -analog of the generalized Apostol type polynomials and the -Stirling numbers of the second kind, the generalized -Bernoulli polynomials of level , the generalized -Apostol type Bernoulli polynomials, the generalized -Apostol type Euler polynomials, the generalized -Apostol type Genocchi polynomials of order and level , and the -Bernstein polynomials.

2. Background and Previous Results

Throughout this paper, we use the following standard notions: , , denotes the set of integers, denotes the set of real numbers, and denotes the set of complex numbers. The -numbers and -factorial numbers are defined, respectively, by El -shifted factorial is defined as The -binomial coefficient is defined by For more information about the -standard definitions and properties see [8, 9, 19].

Furthermore, the -binomial coefficient satisfies the following identity (see [10, p. 483, Eq. ]):The -analog of the function is defined by The -derivative of a function is defined by The -analog of the exponential function is defined in two ways we can see thatTherefore,

For any (see, e.g., [19, p. 76, Eq ]) is called the -gamma function.

The Jackson’s -gamma function is defined by (see [10, p. 490, Eq. ])In (21) we have

For , , , and y the function is defined as (see, e.g., [17, p. 614, Eq. ]) Notice that, when the previous equation is expressed as

Setting y , we can deduceThe -Stirling numbers of the second kind are defined through the following expansion (see, e.g., [20, p. 173, Equ ]):where

Let denote the set of continuous function on . For any , the - is called -Bernstein operator of order for and is defined as (see, e.g., [21, p. 2, Eq ]) where . For , the -Bernstein polynomials of degree or -Bernstein basis are defined by We know that , then and using the identity (13), we have (see [21, p. 6, Eq ])Mahmudov N.I. [22] made a relation between the -Bernstein basis with the -Stirling numbers of the second kind and the -Bernoulli polynomials of order , as follows:

Proposition 1. For a fixed , , and , let be the sequence of generalized -Bernoulli polynomials in of level . Then the following identities are satisfied:
(1) [23, Lemma 10, Eq. (1)](2) [23, Lemma 10, Eq. (2)]Now, setting , we have(3)

Proof (see (36)). Setting , in (7) and using (25), we have Comparing the coefficients of we obtain

Notice that in (36), as we can get (33).

Based on the results of (2), (6), and (7)–(9) and following the methodology given in [12], we consider the following -Mittag-Leffler type function:where , .

3. The Polynomials and Their Properties

Definition 2. Let , , , and ; the -analog of the generalized Apostol type polynomials in , with parameters , order , and level is defined by means of the following generating function, in a suitable neighborhood of ,where when and . The numbers are given by

Notice that Therefore, We will use this notation instead of through the article. By comparing Definition 2 with (7)–(9), we have Therefore, Clearly for , we have For , we have For , we have

Example 3. When , , and we can take . And the numbers are as follows:

Example 4. For any , , , , and

Example 5. For any , , , , and

The following proposition summarizes some properties of the polynomials which are a consequence of (40). Therefore, we will only show the details of its proofs and .

Proposition 6. For a fixed , , let be the sequence of the -analog of the generalized Apostol type polynomials in , parameters , order , and level . Then the following statements hold:
(1) Special values: for every (2) Summation formulas: for every (3) Differential relations: for , fixed and with , we have(4) Integral formulas: for , fixed , we have(5) Addition theorems:Clearly, setting in (64), we have Setting in (67), we obtain (55) Setting and in (64) and (67), respectively, we have (6) If , we have(7) The -analog of the generalized Apostol type polynomials satisfies the following relations:

Proof (see (73)). Considering the expression and using (40), we haveNow, factoring the previous expression, we get Comparing the coefficients of in both sides gives the result.

This completes the proof.

Proof (see (74)). By using the relation (see [23, p.5, Lemma 6])and (40), we have Now, factoring the above expression, we get Comparing the coefficients of in both sides gives the result.

This completes the proof.

Proof (see (75)). Let Using (40) and (79), we have Therefore, we get Notice that Comparing the coefficients of in both sides gives the result.

This completes the proof.

Proof (see (76)). Let Using (40) and (79), we have then, factoring the above equation and using (17), we have Therefore, we get Notice that Comparing the coefficients of in both sides gives the desired result.

This completes the proof.

Remarks 7. Setting , , in (75), we obtainNote that (91) is equivalent to [23, Lemma 6, Eq.2].

Substituting in (75), we obtain

(8) For , , we have

Proof (see (93)). Putting in (75) and using (54), we obtain then

This completes the proof.

4. Some Connection Formulas for the Polynomials

In this section, we introduce some formulas of connection between the -analog of the generalized Apostol type polynomials and the generalized -Bernoulli polynomials of level , the -Stirling numbers of the second kind, the generalized -Apostol type Bernoulli, -Apostol type Euler, -Apostol type Genocchi polynomials of order and level , and the -Bernstein polynomials.

Proposition 8. For , , and , the -analog of the generalized Apostol type polynomials of level is related to the generalized -Bernoulli polynomials of level and the -gamma function

Proof. We only prove (97). Substituting (36) into the right-hand side of (65), we have

Proposition 9. For , , and , the -analog of the generalized Apostol type polynomials is related to the -Stirling numbers of the second kind by means the following identities:

Proof. Substituting (26) into the right-hand side of (55) and (65) gives the results.

Proposition 10. For , , and , the -analog of the generalized Apostol type polynomials is related to the generalized -Apostol-Bernoulli polynomials, the generalized -Apostol-Euler polynomials, and the generalized -Apostol-Genocchi polynomials by means the following identities:

We will only show the details of the demonstrations of (102) and (103).

Proof (see (102)). Using (40) and (79), we have Now, factoring the above equation and using (7), (40), we get Then, we get Comparing the coefficients of in both sides gives the result.

Proof (see (103)). Using (40) and (79), we have Now, factoring the previous equation and using (8), (40), we get Therefore, we get Comparing the coefficients of in both sides gives the result.

Proposition 11. For , , and , the -analog of the generalized Apostol type polynomials is related to -Bernstein basis by means of the following identities:

Proof (see (111)). Substituting (31) into (55) we have

Proof (see (112)). Substituting (31) into (65) we have

Proposition 12. For , , and

Proof (see (115)). The -Bernstein basis is defined by means of following generating function: Using the left-hand side of the previous equation and (40), we have Therefore, we obtain Comparing coefficients the , we obtain

Corollary 13. For , , , and , one has

Proof (see (120)). Substituting (32) into the left-hand side of (115), we obtain the result.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.