New results on the q-generalized Bernoulli polynomials of level m

The q-analogue of the classical Bernoulli numbers and polynomials were initially investigated by Carlitz [2]. More recently, J. Choi, T. Ernst, D. kim, S. Nalci, C.S. Ryoo [3–8] defined the q-Bernoulli polynomials using different methods and studied their properties. There are numerous recent investigations on qgeneralizations of this subject by many others author; see [9–17]. More recently, Mahmudov et al. [18] used the q-Mittag-Le er function E1,m+1(z; q) := zm eq − ∑︀m−1 h=0 zh [h]q ! , m ∈ N,

In the present work, we introduce some algebraic properties from the polynomials given in [18] when α = 1 and λ = 1, called q-generalized Bernoulli B [m−1] n (x; q) of level m, and to research some relations between the q-generalized Bernoulli polynomials of level m and q-gamma function, the q-Stirling numbers of the second kind and the q-Bernstein polynomials.
The paper is organized as follows. Section 2 contains the basic backgrounds about the q-analogue of the generalized Bernoulli polynomials of level m, and some other auxiliary results which we will use throughout the paper. In the Section 3, we introduce some relevant algebraic and differential properties of the qgeneralized Bernoulli polynomials of level m. Finally, in Section 4, we show the corresponding relations between q-generalized Bernoulli polynomials of level m and the q-gamma function, as well as the q-Stirling numbers of the second of the kind and the q-Bernstein polynomials.

Previous definitions and notations
In this paper, we denote by N, N 0 , R, R + , and C the sets of natural, nonnegative integer, real, positive real and complex numbers, respectively. The following q-standard definitions and properties can be found in [19][20][21][22][23]. The q-numbers and q-factorial numbers are defined respectively by The q-shifted factorial is defined as (1 − q j a), a, q ∈ C; |q| < 1.
The q-binomial coefficient is defined by The q-analogue of the function (x + y) n is defined by The q-derivative of a function f (z) is defined by The q-analogue of the exponential function is defined in two ways In this sense, we can see that Therefore, Dq e z q = e z q , Dq E z q = E qz q .
Note that when = 1 the equation above is expressed as From (2.2), setting α = 1 and β = m + 1, we can deduce that The q-Stirling number of the first kind s(n, k)q and the q-Stirling number of the second kind S(n, k)q are the coefficients in the expansions, (see [26, p.173 is called the q-Bernstein operator of order n for f and is defined as (see [15, p.3 Eq. (28)]) where fr = f ([r]q /[n]q). The q-Bernstein polynomials of degree n or a q-Bernstein basis are defined by We know that By using the identity we have Otherwise, setting α = λ = 1 in the equation (1.2), we have the following definition: The first three q-generalized Bernoulli polynomials of level m (cf. [18, p.7]) are Also, the first three q-generalized Bernoulli numbers of level m are Definition 2.4. [14] Let q, α ∈ C, 0 < |q| < 1. The q-Bernoulli polynomials in x, y of order α are defined by means of the generating function where the q-Bernoulli numbers of order α are defined by n (x; q). We will only show in details the proofs to (2), (5) and (7).

Difference equations
Proof. To prove (2), we start with (2.1) and (2.6), from which it follows that and therefore 1 = By comparing coefficients of z n [n]q! , we have By multiplying [n + m] q ! on both sides of the equation above, we have Proof. Proof of (5 By comparing coefficients of z n [n]q! on both sides we obtain the result.
Proof. Proof of (7 Setting x 0 = 0 and x 1 = x in (3.8), we have and so Finally, we get

Some connection formulas for the polynomials B [m−1]
n (x + y; q) From identities (2.4), (2.5) and Proposition 3.1 we can deduce some interesting algebraic relations between the q-generalized Bernoulli polynomials of level m with the q-gamma function, the q-Stirling numbers of the second kind and the q-Bernstein polynomials. .

Corollary 4.2.
For n, j, k ∈ N 0 and m ∈ N, we have Proof. By substituting (3.4) in equation (3.1), we obtain