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

1. Introduction

With the development of q-calculus in the mid-19th century, many authors made generalizations to special functions and polynomial families based on the q-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. (2)])

(1)

Kurt B. [13] did the generalization of the Bernoulli

, Euler

, and Genocchi

polynomials of order α and level m. 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. (12)]):

(2)

to define a extension of the generalized Apostol type polynomials in x, parameters c, a, μ, ν, order α, and level m through the following generating function:

(3)

where

, and

The numbers were given by

(4)

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. (3)]):

(5)

where

, and

. The author named it the unified q-Apostol-Bernoulli, Euler, and Genocchi polynomials of order α and proved some properties for these unification.

On the other hand, in the definition of the q-Mittag-Leffler function when α = 1, β = m + 1, and γ = 1 lead us to (see, [17, p. 614, Eq (1.5)])

(6)

which corresponds to the q-analog of the generating function defined in (1) and with this, new research emerged about other polynomial families based on the q-analogs.

In [18], the authors introduced the generalized q-Apostol-Bernoulli polynomials, the generalized q-Apostol Euler polynomials, and generalized q-Apostol Genocchi polynomials in variable x, y, order α, and level m through the following generating functions, defined in a suitable neighborhood of z = 0 (see, [18, p. 2 Eq (8), (9), (10)])

(7)

(8)

(9)

where

, 0 < |q| < 1, and

Based on the previous result, we focus our attention on a new unification of the q-analog of the generalized Apostol type polynomials of order α and level m, defined in [18], by doing some modifications to the generating function linked to the q-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 q-analogs and some results about q-analogs of the Apostol type polynomials. In Section 3, we introduce the unification q-analog of the generalized Apostol type polynomials in x, y, parameters , order , and level and their algebraic and differential properties. Finally in the Section 4, we show relations between the q-analog of the generalized Apostol type polynomials and the q-Stirling numbers of the second kind, the generalized q-Bernoulli polynomials of level m, the generalized q-Apostol type Bernoulli polynomials, the generalized q-Apostol type Euler polynomials, the generalized q-Apostol type Genocchi polynomials of order α and level m, and the q-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 q-numbers and q-factorial numbers are defined, respectively, by

(10)

El q-shifted factorial is defined as

(11)

The q-binomial coefficient is defined by

(12)

For more information about the q-standard definitions and properties see [8, 9, 19].

Furthermore, the q-binomial coefficient satisfies the following identity (see [10, p. 483, Eq. (41)]):

(13)

The q-analog of the function (x + y)ⁿ is defined by

(14)

The q-derivative of a function f(z) is defined by

(15)

The q-analog of the exponential function is defined in two ways

(16)

we can see that

(17)

(18)

Therefore,

(19)

For any t > 0 (see, e.g., [19, p. 76, Eq (21.6)])

(20)

is called the q-gamma function.

The Jackson’s q-gamma function is defined by (see [10, p. 490, Eq. (2)])

(21)

In (21) we have

(22)

For

, Re(α) > 0, Re(β) > 0, and Re(γ) > 0 y |q| < 1 the function

is defined as (see, e.g., [17, p. 614, Eq. (1.5)])

(23)

Notice that, when γ = 1 the previous equation is expressed as

(24)

Setting α = 1 y β = m + 1, we can deduce

(25)

The q-Stirling numbers of the second kind S(n, k) _q are defined through the following expansion (see, e.g., [20, p. 173, Equ (5.18)]):

(26)

where

(27)

Let C[0,1] denote the set of continuous function on [0,1]. For any f ∈ C[0,1], the q-

is called q-Bernstein operator of order n for f and is defined as (see, e.g., [21, p. 2, Eq (1.1)])

(28)

where f_r = f([r] _q/[n] _q). For

, the q-Bernstein polynomials of degree n or q-Bernstein basis are defined by

(29)

We know that

, then

(30)

and using the identity (13), we have (see [21, p. 6, Eq (2.3)])

(31)

Mahmudov N.I. [22] made a relation between the q-Bernstein basis with the q-Stirling numbers of the second kind and the q-Bernoulli polynomials of order α, α = k as follows:

(32)

Proposition 1. For a fixed , , and 0 < |q| < 1, let be the sequence of generalized q-Bernoulli polynomials in x, y of level m. Then the following identities are satisfied:

(1) [23, Lemma 10, Eq. (1)]

(33)

(2) [23, Lemma 10, Eq. (2)]

(34)

Now, setting k = n − k, we have

(35)

(3)

(36)

Proof (see 36.)Setting α = λ = 1, y = 0 in (7) and using (25), we have

(37)

Comparing the coefficients of zⁿ/[n]_q! we obtain

(38)

Notice that in (36), as Γ_q(n − k + m + 1) = [n − k + m] _q! 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 q-Mittag-Leffler type function:

(39)

where

3. The Polynomials and Their Properties

Definition 2. Let , , , and 0 < |q| < 1; the q-analog of the generalized Apostol type polynomials in x, y, with parameters λ, μ, ν, order α, and level m is defined by means of the following generating function, in a suitable neighborhood of z = 0,

(40)

where |z| < |log_q⁡(−λ)| when

and 1^α≔1. The numbers are given by

(41)

Notice that

(42)

Therefore,

(43)

We will use this notation

instead of

through the article. By comparing Definition 2 with (7)–(9), we have

(44)

Therefore,

(45)

Clearly for m = 1, we have

(46)

For α = 1, we have

(47)

For m = α = 1, we have

(48)

Example 3. When α = 1, μ = 0, and ν = 1 we can take λ = −1.

(49)

And the numbers are as follows:

(50)

Example 4. For any , m = 1, α = 1, μ = 1, and ν = 2

(51)

Example 5. For any , m = 2, α = 1, μ = 3, and ν = 1

(52)

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 (7) and (8).

Proposition 6. For a fixed , 0 < |q| < 1, let be the sequence of the q-analog of the generalized Apostol type polynomials in x, y, parameters , order α, and level m. Then the following statements hold:

(1) Special values: for every

(53)

(54)

(2) Summation formulas: for every

(55)

(56)

(57)

(58)

(3) Differential relations: for , fixed α, λ and with 0 ≤ j ≤ n, we have

(59)

(60)

(4) Integral formulas: for , fixed α, λ, we have

(61)

(62)

(63)

(5) Addition theorems:

(64)

(65)

(66)

(67)

(68)

Clearly, setting x = 0 in (64), we have

(69)

Setting y = 0 in (67), we obtain (55)

(70)

Setting y = 1 and x = 1 in (64) and (67), respectively, we have

(71)

(6) If a ∈ N, we have

(72)

(7) The q-analog of the generalized Apostol type polynomials satisfies the following relations:

(73)

(74)

(75)

(76)

Proof (see (73)). Considering the expression

and using (40), we have

(77)

Now, factoring the previous expression, we get

(78)

Comparing the coefficients of zⁿ/[n] _q! in both sides gives the result.

This completes the proof.

Proof (see (74)). By using the relation (see [23, p.5, Lemma 6])

(79)

and (40), we have

(80)

Now, factoring the above expression, we get

(81)

Comparing the coefficients of zⁿ/[n] _q! in both sides gives the result.

This completes the proof.

Proof (see (75)). Let

(82)

Using (40) and (79), we have

(83)

Therefore, we get

(84)

Notice that

(85)

Comparing the coefficients of zⁿ/[n] _q! in both sides gives the result.

This completes the proof.

Proof (see (76)). Let

(86)

Using (40) and (79), we have

(87)

then, factoring the above equation and using (17), we have

(88)

Therefore, we get

(89)

Notice that

(90)

Comparing the coefficients of zⁿ/[n] _q! in both sides gives the desired result.

This completes the proof.

Remarks 7. Setting μ = 1, ν = 0, λ = 1 in (75), we obtain

(91)

Note that (91) is equivalent to [23, Lemma 6, Eq.2].

Substituting ν = 0 in (75), we obtain

(92)

(8) For

, n ≥ νm, we have

(93)

Proof (see (93)). Putting α = 1 in (75) and using (54), we obtain

(94)

then

(95)

This completes the proof.

4. Some Connection Formulas for the Polynomials

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

Proposition 8. For , , and 0 < |q| < 1, the q-analog of the generalized Apostol type polynomials of level m is related to the generalized q-Bernoulli polynomials of level m and the q-gamma function

(96)

(97)

(98)

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

(99)

Proposition 9. For , , and 0 < |q| < 1, the q-analog of the generalized Apostol type polynomials is related to the q-Stirling numbers of the second kind S(n, k; q) by means the following identities:

(100)

(101)

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

Proposition 10. For , , and 0 < |q| < 1, the q-analog of the generalized Apostol type polynomials is related to the generalized q-Apostol-Bernoulli polynomials, the generalized q-Apostol-Euler polynomials, and the generalized q-Apostol-Genocchi polynomials by means the following identities:

(102)

(103)

(104)

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

Proof (see 102.)Using (40) and (79), we have

(105)

Now, factoring the above equation and using (7), (40), we get

(106)

Then, we get

(107)

Comparing the coefficients of zⁿ/[n] _q! in both sides gives the result.

Proof (see 103.)Using (40) and (79), we have

(108)

Now, factoring the previous equation and using (8), (40), we get

(109)

Therefore, we get

(110)

Comparing the coefficients of zⁿ/[n] _q! in both sides gives the result.

Proposition 11. For , , and 0 < |q| < 1, the q-analog of the generalized Apostol type polynomials is related to q-Bernstein basis by means of the following identities:

(111)

(112)

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

(113)

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

(114)

Proposition 12. For , , and 0 < |q| < 1

(115)

Proof (see 115.)The q-Bernstein basis is defined by means of following generating function:

(116)

Using the left-hand side of the previous equation and (40), we have

(117)

Therefore, we obtain

(118)

Comparing coefficients the zⁿ/[n] _q!, we obtain

(119)

Corollary 13. For , 0 ≤ k ≤ j ≤ n, , and 0 < |q| < 1, one has

(120)

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Open Research

Data Availability

No data were used to support this study.

References

1 Carlitz L., q-Bernoulli numbers and polynomials, Duke Mathematical Journal. (1948) 15, 987–1000, https://doi.org/10.1215/S0012-7094-48-01588-9, MR0027288.
10.1215/S0012-7094-48-01588-9
Web of Science® Google Scholar
2 Cieslinski J. L., Improved q-exponential and q-trigonometric functions, Applied Mathematics Letters. (2011) 24, no. 12, 2110–2114.
10.1016/j.aml.2011.06.009
Web of Science® Google Scholar
3 Ernst T., q-Bernoulli and q-Euler polynomials, an umbral approach, International Journal of Difference Equations. (2006) 1, no. 1, 31–80, MR2296498.
Google Scholar
4 Garg M. and Alha S., Una nueva clase de polinomios q-Apostol-Bernoulli de orden a /A new class of q-Apostol-Bernoulli polynomials of order a, Revista Tecnocientífica URU. (2014) 6, 67–76.
Google Scholar
5 Koornwinder T. H., Special functions and q-commuting variables, https://arxiv.org/abs/q-alg/9608008, 1996.
Google Scholar
6 Luo Q. M., q-Extensions of Some Results Involving the Luo-Srivastava Generalizations, Filomat. (2014) 28, no. 2, 329–351.
10.2298/FIL1402329L
Web of Science® Google Scholar
7 Luo Q. M., Some results for the q-Bernoulli andq-Euler polynomials, Journal of Mathematical Analysis and Applications. (2010) 363, no. 1, 7–18.
10.1016/j.jmaa.2009.07.042
Web of Science® Google Scholar
8 Ernst T., The History of q-Calculus And A New Method, 2000, Department of Mathematics, Uppsala University, Uppsala, Sweden.
Google Scholar
9 Gasper G. and Rahman M., Basic Hypergeometric Series, 2004, 96, Cambridge university press, Cambridge, UK.
10.1017/CBO9780511526251
Google Scholar
10 Srivastava H. M. and Junesang Ch., Zeta And q-Zeta Functions And Associated Series And Integrals, 2012, Editorial Elsevier, Boston, Massachusetts.
Google Scholar
11 Natalini P. and Bernardini A., A generalization of the Bernoulli polynomials, Journal of Applied Mathematics. (2003) no. 3, 155–163, https://doi.org/10.1155/S1110757X03204101, MR1982355, Zbl1019.33011, 2-s2.0-4744345928.
10.1155/S1110757X03204101
Google Scholar
12 Hernández-Llanos P., Quintana Y., and Urieles A., About Extensions of Generalized Apostol-Type Polynomials, Results in Mathematics. (2015) 68, no. 1-2, 203–225, 2-s2.0-84940590890, https://doi.org/10.1007/s00025-014-0430-2, Zbl1335.11014.
10.1007/s00025-014-0430-2
Web of Science® Google Scholar
13 Kurt B., Some Relationships between the Generalized Apostol-Bernoulli and Apostol-Euler Polynomials, Turkish Journal of Analysis and Number Theory. (2013) 1, no. 1, 54–58, https://doi.org/10.12691/tjant-1-1-11.
10.12691/tjant-1-1-11
Google Scholar
14 Tremblay R., Gaboury S., and Fugere J., A further generalization of Apostol-Bernoulli polynomials and related polynomials, Honam Mathematical Journal. (2012) 34, no. 3, 311–326.
10.5831/HMJ.2012.34.3.311
Google Scholar
15 Tremblay R., Gaboury S., and Fugere B. J., A new class of generalized Apostol–Bernoulli polynomials and some analogues of the Srivastava–Pintér addition theorem, Applied Mathematics Letters. (2011) 24, no. 11, 1888–1893.
10.1016/j.aml.2011.05.012
Web of Science® Google Scholar
16 Kurt B., Notes on Unified q-Apostol-Type Polynomials, Filomat. (2016) 30, no. 4, 921–927.
10.2298/FIL1604921K
Web of Science® Google Scholar
17 Sharma S. K. and Jain R., On some properties of generalized q-Mittag leffler, Mathematica Aeterna. (2014) 4, no. 6, 613–619.
Google Scholar
18 Mahmudov N. I. and Keleshteri M. E., q-extensions for the Apostol type polynomials, Journal of Applied Mathematics. (2014) Art. ID 868167, 8, https://doi.org/10.1155/2014/868167, MR3219435.
10.1155/2014/868167
Google Scholar
19 Kac V. and Cheung P., Quantum Calculus, 2001, Science and Business Media, Berlin, Germany.
Google Scholar
20 Ernst T., A Comprehensive Treatment of q-Calculus, 2012, Springer Science and Business Media, Berlin, Germany.
10.1007/978-3-0348-0431-8
Google Scholar
21 Oruç H. and Phillips G. M., Bernstein polynomials and Bézier curves, Journal of Computational and Applied Mathematics. (2003) 151, no. 1, 1–12.
10.1016/S0377-0427(02)00733-1
Web of Science® Google Scholar
22 Mahmudov N. I., On a class of q-Bernoulli andq-Euler polynomials, Advances in Difference Equations. (2013) 1.
Google Scholar
23 Mahmudov N. I. and Keleshteri M. E., On a class of generalized q-Bernoulli and q-Euler polynomials, Advances in Difference Equations. (2013) 1.
Google Scholar

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An Extended Generalized q-Extensions for the Apostol Type Polynomials

Abstract

1. Introduction

2. Background and Previous Results

3. The Polynomials and Their Properties

4. Some Connection Formulas for the Polynomials

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

References

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An Extended Generalized q-Extensions for the Apostol Type Polynomials

Abstract

1. Introduction

2. Background and Previous Results

3. The Polynomials and Their Properties

4. Some Connection Formulas for the Polynomials

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

References

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