Abstract and Applied Analysis
Volume 2018 (2018), Article ID 2937950, 13 pages
https://doi.org/10.1155/2018/2937950

Research Article

An Extended Generalized -Extensions for the Apostol Type Polynomials

Letelier Castilla,¹ William Ramírez,² and Alejandro Urieles³

¹Universidad del Atlántico, Barranquilla, Colombia
²GICBAS, Universidad de la Costa, Barranquilla, Colombia
³Programa de Matemática, Universidad del Atlántico, Km 7, Vía Pto. Colombia, Barranquilla, Colombia

Correspondence should be addressed to Alejandro Urieles; alejandrourieles@mail.uniatlantico.edu.co

Received 5 February 2018; Revised 3 May 2018; Accepted 22 May 2018; Published 2 July 2018

Academic Editor: Allan Peterson

Copyright © 2018 Letelier Castilla et al. 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.

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 that

Therefore,

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 deduce

The -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 have

Now, 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 obtain

Note 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.

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