Approximation properties of q-Bernoulli polynomials

We study the q-analogue of Euler-Maclaurin formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials is discussed. We estimate the suitable functions as a combination of truncated series of q-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches, first solving differential equations by estimating functions and second we may apply these techniques for operator theory.


Introduction
The present study has sough to investigate the approximation of suitable function f (x) as a linear combination of q−Bernoulli polynomials. This study by using q−operators through a parallel way has achived to the kind of Euler expansion for f (x), and expanded the function in terms of q−Bernoulli polynomials. This expansion offers a proper tool to solve q-difference equations or normal differential equation as well. There are many approaches to approximate the capable functions. According to properties of q-functions many q-function have been used in order to approximate a suitable function.For example, at [28], some identities and formulae for the q-Bernstein basis function, including the partition of unity property, formulae for representing the monomials were studied. In addition, a kind of approximation of a function in terms of Bernoulli polynomials are used in several approaches for solving differential equations, such as [7,20,21]. This paper gives conditions to approximate capable functions as a linear combination of q-Bernoulli polynomials as well as related examples. Also, introduces the new q−operator to reach the q−analogue of Euler-Maclaurin formula.
This study, first, introduces some q-calculus concepts. There are several types of q−Bernoulli polynomials and numbers that can be generated by different q−exponential functions. Carlitz [2] is pionier of introducing q-analogue of the Bernoulli numbers, he applied a sequence {β m } m≥0 in the middle of the 20 th century; First, this study makes an assumption that |q| < 1, and this assumption is going to apply in the rest of the paper as well. If q tends to one from a left side the ordinary form would be reached.
Since Carlitz, there have been many distinct q-analogue of Bernoulli numbers arising from varying motivations. In [3], they used improved q−exponential functions to introduce a new class of q−Bernoulli polynomials. In addition, they investigate some properties of these q-Bernoulli polynomials.
Let us introduce a class of q−Bernoulli polynomials in a form of generating function as follows; where [n] q is q−number and q-numbers factorial is defined by; The q-shifted factorial and q-polynomial coefficient are defined by the following expressions respectively; q-standard terminology and notation can be found at [1] and [6]. We call β n,q Bernoulli number and β n,q = β n,q (0). In addition, q-analogue of (x − a) n is defined as; In the standard approach to the q−calculus, two exponential function are used.These q−exponentials are defined by; q−shifted factorial can be expressed by Heine's binomial formula as follow; (a; q) n = n k=0 n k q q Let for some 0 ≤ α < 1, the function |f (x)x α | is bounded on the interval (0, A], then Jakson integral defines as [1]; Above expression converges to a function F (x) on (0, A] , which is a q−antiderivative of f (x). Suppose 0 < a < b, the definite q−integral is defined as We need to apply some properties of the q-Bernoulli polynomials to prepare the approximation conditions. These properties are listed below as a lemma; Lemma 1 Following statements holds true: Proof. taking q−derivative from both side of (2) to prove part (a). If we apply Cauchy product for generating formula (4), it leads to (b). Put x = 0 at part (b) and change boundary of summation to reach (c). If we take Jackson integral directly from β n,q (t) with the aid of part (c), we can reach (d).
A few numbers of these polynomials and related q−Bernoulli numbers can be expressed as follows Definition 2 q−Bernoulli polynomials of two variables are defined as a generating function as follow A simple calculation of generating function leads us to The LHS can be written as By comparing the n th coefficients, we drive to difference equations as follow If we put x = 0, then β n,q (0, 1) − β n,q (0) = δ n,1 .

q-analogue of Euler-Maclaurin formula
This section introduces q−operator to find q−analogue of Euler-Maclaurin formula. The q−analogue of Euler-Maclaurin formula has been studied in [29]. They applied q-integral by parts to reach their formula. We can not apply that approach to approximation, because it was written in term of Moreover, the errors has not been studied and our q−operator which is totally new, leads us to the more applicable function. That is why, this study situate q−Taylor theorem first [6] Theorem 3 If the function f (x) is capable of expansion as a convergent power series and if q = root of unity, then , for x = 0, we can define it at x = 0 as a normal derivative.

Definition 4
We define H q and D h operators as follow In this definition we assume that n ∈ N and the functions and values are well-defined (h = 0, q = 1) . The fist equality is hold because of Heine's binomial formula.
For a long time, mathematician have worked on the area of operators and they solved several types of differential equations by using shifted-operators. The H−operators rules like a bridge between the ordinary expansions and q-analogue of these expansions. We may rewrite several formulae of these area to the form of q−calculus such as [23,24,25,26,27]. Actually we may write q−expansion of these functions. In a letter, Bernoulli concerned the importence of this expansion for Leibnitz by these words" Nothing is more elegent than the agreement, which you have observed between the numerical power of the binomial and differetial expansions, there is no doubt that something is hidden there" [26] where we define h-integral as follows Now in the aid of q−Taylor expansion, we may write F (x + h) as follows; Here, e q is q−shifted operator and can be expressed as expansion of D q and H q . We may assume that D h (F (x)) = f (x) and then Since the Jackson integral is q−antiderivative, we have And in the aid of (2), the left hand side of the equation can be expressed as a q-Bernoulli numbers, therefore Thus we can state the following q−analogue of Euler-Maclaurian formula

Theorem 6
If the function f (x) is capable of expansion as a convergent power series and f (x) decrease so rapidly with x such that all normal derivatives approach zero as x → ∞, then we can express the series of function f (x) as follow Proof. In the aid of theorem (5) and (18), If we suppose that h = 1 and b − a ∈ N, we have Let f (x) decrease so rapidly with x then D q f (x) can be estimated by −f (x) x(q−1) , tend x to infinity and use L'Hopital to reach −f ′ (x) (q−1) .Now If normal derivatives of f (x) is tend to zero, then q−derivatives is also tending zero. The same discussion make D n−1 Example 7 Let f (x) = x s where s is a positive integer, then we can apply this function at (22) where a = 0 This relation shows the sum of power in the combination of q−Bernoulli numbers. when q → 1 from the right side, we have an ordinary form of this relation. For the another forms of sum of power, see [22] Example 8 Let f (x) = e −x q = 1 E x q then this function decreases so rapidly with x such that all normal derivatives approach zero as x → ∞. For comfirming this, let us mention that E x q , for some fixed 0 < |q| < 1 and | − x| < 1 |1−q| is increasing rapidly, since d (2n)! , where β 2n is a normal Bernoulli numbers that is generated by −1 e −1 −1 .

approximation by q-Bernoulli polynomial
We know that the class of q-Bernoulli polynomials are not in a form of orthogonal polynomials. In addition, we may apply Gram-Schmit algorithm to make these polynomials orthonormal then, according to Theorem 8.11 [19] we have the best approximation in the form of Fourier series. Now, Instead of using that algorithm, we apply properties of lemma (1) to achieve an approximation. Let H = L 2 be the Hilbert Space [18], then β n,q (x) ∈ H for n = 0, ..., N and Y = Span {β 0,q (x), β 1,q (x), ..., β N,q (x)} is finite dimensional vector subspace of H. Unique best approximation for any arbitrary elements of H like h, is h ∈ Y such that for any y ∈ Y the inequality h − h . Following proposition determine the coefficient of q−Bernoulli polynomials, when we estimate any f ∈ L 2 q [0, 1] by truncated q−Bernoulli series. In fact, in order to see how well a certain partial sum approximates the actual value, we would like to drive a formula similar to (19), but with the infinite sum on the right hand side replaced by the N th partial sum S N , plus an additional term R N.
Sippose a ∈ Z and b = a + 1.Consider the N th partial sum Let g(x, y) = βN,q(x,y) [N ] q ! , then in the aid of lemma (1) we have D n−k q g(x, y) = β k,q (x,y) [k] q ! and we have By taking q−derivative of the summation, we may rewrite this partial sum by integral presentation, Moreover this relation give a boundary for R N (x).
Proposition 9 Let f ∈ L 2 q [0, 1] and be estimated by truncated q−Bernoulli series N n=0 C n β n,q (x). Then C n coefficients for n = 0, 1, ..., N can be calculated as an integral form C n = 1 In addition we can write f (x) in the following form Moreover R q,n (x) as a reminder part is bounded by 2 n In fact we approximate f (x) as a linear combinations of q−Bernoulli polynomials and we assume that f (x) ≃ N n=0 C n β n,q (x), so take the Jackson integral from both sides lead us to In the aid of Lemma 1 part (d) all the terms at the right side except the first one has to be zero. we calculate β 0,q (x) = 1, so C 0 = 1  Repeating this procedure n-times, yields the form of C n as we mention it at theorem. We mention that if x ∈ [0, 1] and 0 < |q| < 1, then H n q (x) for h = 1 is bounded by 2 n .In addition, the reminder part can be presented by integral forms and this boundary can be found easily.

Further works
In this paper, we introduced a proper tool to approximate a given capable function by combinations of q-Bernoulli polynomials. In spite of a lot of investigations on q-Bernoulli polynomials, study of approximation properties of the q-Bernoulli polynomials are not studied. We may apply these results to solve q-difference equation, q-analogue of the things that is done in [20] or [21]. The techniques of H-operator can be applied in operator theory.

conflict of interest
The authors declare that there is no conflict of interest regarding the publication of this paper