Approximation Properties of -Bernoulli Polynomials
We study the -analogue of Euler-Maclaurin formula and by introducing a new -operator we drive to this form. Moreover, approximation properties of -Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of -Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.
The present study has sought to investigate the approximation of suitable function as a linear combination of -Bernoulli polynomials. This study applies q-operator to expand a function in terms of q-Bernoulli polynomials. In addition, this method leads us to q-analogue of Euler expansion. This expansion offers a proper tool to solve -difference equations or normal differential equation as well. There are many approaches to approximate the capable functions. According to properties of -functions many -functions have been used in order to approximate a suitable function. For example, at , some identities and formulae for the -Bernstein basis function, including the partition of unity property, and formulae for representing the monomials were studied. In addition, a kind of approximation of a function in terms of Bernoulli polynomials is used in several approaches for solving differential equations, such as [2–4]. This paper gives conditions to approximate capable functions as a linear combination of -Bernoulli polynomials as well as related examples. Also, it introduces the new -operator to reach the -analogue of Euler-Maclaurin formula.
This study, first, introduces some -calculus concepts. There are several types of -Bernoulli polynomials and numbers that can be generated by different -exponential functions. Carlitz  is pioneer of introducing -analogue of the Bernoulli numbers; he applied a sequence in the middle of the 20th century;
First, this study makes an assumption that , and this assumption is going to apply in the rest of the paper as well. If tends to one from a left side the ordinary form would be reached.
Since Carlitz, there have been many distinct -analogues of Bernoulli numbers arising from varying motivations. In , the authors used improved -exponential functions to introduce a new class of -Bernoulli polynomials. In addition, they investigate some properties of these -Bernoulli polynomials.
Let us introduce a class of -Bernoulli polynomials in a form of generating function as follows:where is -number and -numbers factorial is defined byThe -shifted factorial for the case , for natural , and for infinity case and -polynomial coefficient are defined by the following expressions, respectively:-standard terminology and notation can be found at  and . We call Bernoulli number and . In addition, -analogue of is defined as
In the standard approach to the -calculus, two exponential functions are used. These -exponentials are defined by
-shifted factorial can be expressed by Heine’s binomial formula as follows:
For some , the function is bounded on the interval , then Jackson integral is defined as 
The above expression converges to a function on , which is a -antiderivative of . Suppose ; the definite -integral is defined as
We need to apply some properties of the -Bernoulli polynomials to prepare the approximation conditions. These properties are listed below as a lemma.
Lemma 1. The following statements hold true:
Proof. Taking -derivative from both sides of (2) to prove part (a). If we apply Cauchy product for generating formula (11), it leads to (b). Put at part (b) and change boundary of summation to reach (c). If we take Jackson integral directly from with the aid of part (c), we can reach (d).
A few numbers of these polynomials and related -Bernoulli numbers can be expressed as follows:
Definition 2. -Bernoulli polynomials of two variables are defined as a generating function as follows:A simple calculation of generating function leads us toThe LHS can be written asBy comparing the th coefficients, we drive to difference equations as follows:If we put , then
2. -Analogue of Euler-Maclaurin Formula
This section introduces -operator to find -analogue of Euler-Maclaurin formula. The -analogue of Euler-Maclaurin formula has been studied in . The authors of  applied q-integral by parts to reach q-analogue of Euler-Maclaurin formula. We can not apply that approach to approximation, because it was written in terms of . Moreover, the errors have not been studied and our -operator, which is totally new, leads us to the more applicable function. That is why, this study situates -Taylor theorem first .
Theorem 3. If the function is capable of expansion as a convergent power series and if root of unity, thenwhere is -derivative of , and for , we can define it at as a normal derivative.
Definition 4. One defines and operators as follows:In this definition, we assume that and the functions and values are well defined The first equality holds because of Heine’s binomial formula.
For a long time, mathematicians have worked on the area of operators and they solved several types of differential equations by using shifted-operators. The -operator rules like a bridge between the ordinary expansions and -analogue of these expansions. We may rewrite several formulae of these areas to the form of -calculus such as [10–14]. Actually we may write -expansion of these functions. In a letter, Bernoulli concerned the importance of this expansion for Leibnitz by these words “Nothing is more elegant than the agreement, which you have observed between the numerical power of the binomial and differential expansions, there is no doubt that something is hidden there” .
Theorem 5 (fundamental theorem of -calculus). If is an -derivative of and , one has where we define -integral as follows:
Now in the aid of -Taylor expansion, we may write as follows:
Here, is -shifted operator and can be expressed as expansion of and We may assume that and then
Since the Jackson integral is -antiderivative, we have
And in the aid of (2), the left hand side of the equation can be expressed as a -Bernoulli numbers; therefore
Thus we can state the following -analogue of Euler-Maclaurin formula.
Theorem 6. If the function is capable of expansion as a convergent power series and decreases so rapidly with such that all normal derivatives approach zero as , then one can express the series of function as follows:
Proof. In the aid of Theorem (12) and (24), if we suppose that and , we haveLet decrease so rapidly with , since , then by applying the limits when tends to infinity, reach zero. Now if normal derivatives of tend to zero, then -derivatives are also tending to zero. The same discussion makes for big enough . Therefore
Example 7. Let , where is a positive integer, then we can apply this function at (28), where and This relation shows the sum of power in the combination of -Bernoulli numbers. When from the right side, we have an ordinary form of this relation. For the another forms of sum of power, see .
Example 8. Let , then this function decreases so rapidly with such that all normal derivatives approach zero as For confirming this, let us mention that , for some fixed and , is increasing rapidly, since Moreover, this is -analogue of , where is a normal Bernoulli number that is generated by
3. Approximation by -Bernoulli Polynomial
We know that the class of -Bernoulli polynomials are not in a form of orthogonal polynomials. In addition, we may apply Gram-Schmidt algorithm to make these polynomials orthonormal, then, according to Theorem , 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 be the Hilbert space , then for and is finite dimensional vector subspace of . Unique best approximation for any arbitrary elements of like is such that for any the inequality , where the norm is defined by Following proposition determine the coefficient of -Bernoulli polynomials, when we estimate any by truncated -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 (25), but with the infinite sum on the right hand side repla by the th partial sum , plus an additional term .
Suppose and Consider the th partial sum
Let , then in the aid of Lemma 1 we have and we have
By taking -derivative of the summation, we may rewrite this partial sum by integral presentation. Moreover this relation gives a boundary for
Proposition 9. Let and be estimated by truncated -Bernoulli series Then coefficients for can be calculated as an integral form . In addition one can write in the following form:Moreover as a reminder part is bounded by .
Proof. In fact we approximate as a linear combination of -Bernoulli polynomials and we assume that , so taking the Jackson integral from both sides leads us toIn the aid of Lemma 1 part (d), all the terms at the right side except the first one have to be zero. We calculate , so Using part (a) of that lemma for -derivative of -Bernoulli polynomial gathered by taking -derivative of leads toRepeating this procedure -times yields the form of as we mention it at theorem. We mention that if and , then for is bounded by In addition, the reminder part can be presented by integral forms and this boundary can be found easily.
3.1. Further Works
In this paper, we introduced a proper tool to approximate a given capable function by combinations of -Bernoulli polynomials. In spite of a lot of investigations on -Bernoulli polynomials, approximation properties of the -Bernoulli polynomials are not studied. We may apply these results to solve -difference equation, -analogue of the things that is done in  or . The techniques of -operator can be applied in operator theory.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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