Applications of q-Umbral Calculus to Modified Apostol Type q-Bernoulli Polynomials

Corresponding author: Serkan Araci Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey Email: mtsrkn@hotmail.com Abstract: This article aims to identify the generating function of modified Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modified Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are qAppell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modified q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.


Introduction
Throughout this paper, we make use of the following standard notations: . Also, as usual, ℤ denotes the set of integers, ℝ denotes the set of real numbers and ℂ denotes the set of complex numbers.
We now begin with the fundamental properties of q-calculus. Let q be chosen as a fixed real number between 0 and 1. The q-analogue of any number n is given by: These fundamental properties of q-calculus listed above are taken from the book (Kac and Cheung, 2002).
By using an exponential function e q (x), Kupershmidt (2005) defined the following q-Bernoulli polynomials: e xt e t n ∞ = = − ∑ In the case x = 0, B n,q (0) = B n,q means the n-th q-Bernoulli number.
Very recently, Kurt (2016) defined Apostol type q-Bernoulli polynomials of order α by making use of the following generating function: x y e xt E yt e t n α α λ λ where, λ ∈ ℂ and 0 α ∈ ℕ . In this study, we will study on the following polynomial ( ) ( ) given by special cases α = 1 and y = 0 in (1.2): When q → 1 in (1.3), it reduces to Apostol-Bernoulli polynomials (Choi et al., 2008;Luo and Srivastava, 2006).
We now review briefly the concept of q-umbral calculus. For the properties of q-umbral calculus, we refer the reader to see the references (Araci et al., 2007;Choi et al., 2008;Kac and Cheung, 2002;Kim and Kim, 2014a;Kim et al., 2013;Mahmudov and Keleshteri, 2013;Roman, 1985).
Let ℂ be a field of characteristic zero and let F be the set of all formal power series in the variable t over ℂ with: Let P be the algebra of polynomials in the single variable x over the field complex numbers and let * P be the vector space of all linear functionals on P . In the q-Umbral calculus, 〈L|p(x)〉 means the action of a linear functional L on the polynomial p(x). This operator has a linear property on * P given by: The formal power series: defines a linear functional on P by setting: Taking f(t) = t k in Equation 1.4 and 1.5 gives: Actually, any linear functional L in * P has the form (1.4). That is, since: and so as linear functionals L = f L (t). Moreover, the map L → f L (t) is a vector space isomorphism from * P onto F. Henceforth, F will denote both the algebra of formal power series in t and the vector space of all linear functionals on P and so an element f(t) of F will be thought of as both a formal power series and a linear functional. From (1.5), we have: e yt x y = and so: The order o(f(t)) of a power series f(t) is the smallest integer k for which the coefficient of t k does not vanish. If o(f(t)) = 0, then f(t) is called an invertible series. A series f(t) for which o(f(t)) = 1 will be called a delta series (Araci et al., 2007;Choi et al., 2008;Kac and Cheung, 2002;Kim and Kim, 2014a;Kim et al., 2013;Mahmudov and Keleshteri, 2013;Roman, 1985).
If f 1 (t), ..., f m (t) are in F, then: We use the notation t k for the k-th q-derivative operator on P as follows: for all polynomials p(x). Notice that for all f(t) in F and for all polynomials p(x): Using (1.7), we obtain: Thus, from (1.8), we note that: Let f(t) ∈ F be a delta series and let g(t) ∈ F be an invertible series. Then there exists a unique sequence s n (x) of polynomials satisfying the following property: which is called an orthogonality condition for any qsheffer sequence, cf. (Araci et al., 2007;Choi et al., 2008;Kac and Cheung, 2002;Kim and Kim, 2014a;Kim et al., 2013;Mahmudov and Keleshteri, 2013;Roman, 1985). The sequence s n (x) is called the q-Sheffer sequence for the pair of (g(t), f(t)), or this s n (x) is q-Sheffer for (g(t), f(t)), which is denoted by s n (x) ∼ (g(t), f(t)).
Let s n (x) be q-Sheffer for (g(t), f(t)). Then for any h(t) in F and for any polynomial p(x), we have: An important property for the q-Sheffer sequence s n (x) having (g(t), t) is the q-Appell sequence. It is also called q-Appell for g(t) with the following consequence: Further important property for q-Sheffer sequence s n (x) is as follows: For having information about the properties of qumbral theory (Araci et al., 2007;Choi et al., 2008;Kac and Cheung, 2002;Kim and Kim, 2014a;Kim et al., 2013;Mahmudov and Keleshteri, 2013;Roman, 1985) and cited references therein.

Modified Apostol Type q-Bernoulli Numbers and Polynomials
Recall from (1.3) that: Taking t → 0 on the above gives B 0,q (x, λ) = 0. This shows that the generating function of these polynomials is not invertible. Therefore, we need to modify slightly Equation (2.1) as follows: This modification yields to being invertible for generating function of modified Apostol type q-Bernoulli polynomials. As a traditional for some special polynomials to be a number, in the case when x = 0, B λ is called the modified Apostol type n-th q-Bernoulli number. Now we list some properties of modified Apostol type q-Bernoulli polynomials as follows.
From (2.2), we obtain: , n n n k k n q k q n k q k k q q By (2.2), the modified Apostol type q-Bernoulli numbers can be found by means of the following recurrence relation:  , n q B x λ is q-Appell for λe q (t)-1.
We now have the following theorem.

Theorem 1
Let ( ) p x ∈ P . We have: Thus we have the following interesting property for modified Apostol type q-Bernoulli numbers derived from Theorem 1 for n ≥ 0: The following is an immediate result emerging from (1.10) and (2.5) that: By choosing suitable polynomials p(x), one can derive some interesting results. So we omit to give examples and so we now take care of a fundamental property in qumbral theory which is stated below by Theorem 2.

Theorem 2
Let n be nonnegative integer. Then we have: Thus, by applying (2.8), we get: Comparing Equation (2.8) with Equation (2.9), we complete the proof of this theorem. The following theorem is useful to derive any polynomial as a linear combination of modified Apostol type q-Bernoulli polynomials.

Theorem 3
For q(x) ∈ P n , let:

Proof
It follows from (1.9) that: (2.10) We now consider the following sets of polynomials of degree less than or equal to n: For ( ) n q x ∈ P , we further consider that: Combining (2.10) with (2.11), it becomes: Thus, from (2.12), we have: Thus the proof is completed. When we choose q(x) = E n,q (x), we have the following corollary which is given by its proof.
n n q n q n q n q n q k q k q k n q n q q n n k q k q k q Recall from (1.2) that Apostol type q-Bernoulli polynomials of order r are given by the following generating function, for y = 0 (Kurt, 2016): where, t ∈ ℂ and r ∈ ℕ. If t approaches to 0 on the above, it yields to ( ) ( ) 0, , q B x α λ = 0, which means that the generating function of ( ) ( ) , , n q B x α λ is not invertible. So, we need to modify slightly Equation (2.1), as follows: which implies an invertible since: Here we find that:

Theorem 4
Let n be nonnegative integer. Then we have: where the coefficient b k,q is given by: From the Equation (2.19) to (2.21), we get the following theorem.