# Approximation Properties ofq-Bernoulli Polynomials

Authors: M. Momenzadeh; I. Y. Kakangi
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

We study theq-analogue of Euler-Maclaurin formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials are 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 we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.


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## Body

## 1. Introduction

The present study has sought to investigate the approximation of suitable functionf(x) as a linear combination of q-Bernoulli polynomials. This study appliesq-operator to expand a function in terms ofq-Bernoulli polynomials. In addition, this method leads us toq-analogue of Euler expansion. 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-functions have been used in order to approximate a suitable function. For example, at [1], some identities and formulae for the q-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 q-Bernoulli polynomials as well as related examples. Also, it introduces the new q-operator to reach the q-analogue of Euler-Maclaurin formula.This study, first, introduces someq-calculus concepts. There are several types of q-Bernoulli polynomials and numbers that can be generated by different q-exponential functions. Carlitz [5] is pioneer of introducing q-analogue of the Bernoulli numbers; he applied a sequence βmm≥0 in the middle of the 20th century;(1)∑k=0mmkβkqk+1-βm=1,m=1,0,m>1.First, this study makes an assumption thatq<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 distinctq-analogues of Bernoulli numbers arising from varying motivations. In [6], the authors 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 ofq-Bernoulli polynomials in a form of generating function as follows:(2)zeqzteqz-1=∑n=0∞βn,qtznnq!,z<2π,where [n]q is q-number and q-numbers factorial is defined by(3)aq=1-qa1-qq≠1;0!=1;n!=nn-1!n∈N,a∈C.The q-shifted factorial for the case n=0, for natural n, and for infinity case and q-polynomial coefficient are defined by the following expressions, respectively:(4)a;q0=1,a;qn=∏j=0n-11-qja,n∈N,a;q∞=∏j=0∞1-qja,q<1,a∈C;nkq=q;qnq;qn-kq;qk,q-standard terminology and notation can be found at [7] and [8]. We call βn,q Bernoulli number and βn,q=βn,q(0). In addition, q-analogue of x-an is defined as(5)x-aqn=1,n=0,x-ax-aq⋯x-aqn-1,n≠0.In the standard approach to theq-calculus, two exponential functions are used. These q-exponentials are defined by(6)eqz=∑n=0∞znnq!=∏k=0∞11-1-qqkz,0<q<1,z<11-q,Eqz=e1/qz=∑n=0∞q1/2nn-1znnq!=∏k=0∞1+1-qqkz,0<q<1,z∈C.q-shifted factorial can be expressed by Heine’s binomial formula as follows:(7)a;qn=∑k=0nnkqqkk-1/2-1kak.For some0≤α<1, the function |f(x)xα| is bounded on the interval 0,A, then Jackson integral is defined as [7](8)∫fxdqx=1-qx∑i=0∞qifqix.The above expression converges to a functionF(x) on 0,A, which is a q-antiderivative of f(x). Suppose 0<a<b; the definite q-integral is defined as(9)∫0bfxdqx=1-qb∑i=0∞qifqib,∫abfxdqx=∫0bfxdqx-∫0afxdqx.We need to apply some properties of theq-Bernoulli polynomials to prepare the approximation conditions. These properties are listed below as a lemma.Lemma 1.
The following statements hold true:(10)aDqβn,qt=nqβn-1,qt,(11)bβn,qt=∑k=0nnkqβk,qtn-k,(12)c∑k=0nnkqβk,qn-k+1q=δ0,nwhereδiskroneckerdelta,(13)d∫01βn,qtdqt=0n≠0.Proof.
Takingq-derivative from both sides of (2) to prove part (a). If we apply Cauchy product for generating formula (11), 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 βn,q(t) with the aid of part (c), we can reach (d).A few numbers of these polynomials and relatedq-Bernoulli numbers can be expressed as follows:(14)β0,q=1,β0,qt=1,β1,q=-12q,β1,qt=t-12q,β2,q=q23q!,β2,qt=t2-t+q23q!,β3,q=q31-q5q+4q-1,β3,qt=t3-3q2qt2+q22qt+q31-q5q+4q-1.Definition 2.
q-Bernoulli polynomials of two variables are defined as a generating function as follows:(15)teqxteqyteqt-1=∑n=0∞βn,qx,ytnnq!,t<2π.
A simple calculation of generating function leads us to(16)teqxt+teqt-1eqtx=teqxteqt-1eqt=∑n=0∞βn,qx,1tnnq!,t<2π.
The LHS can be written as(17)∑n=0∞tn+1xnnq!+∑n=0∞βn,qxtnnq!.
By comparing thenth coefficients, we drive to difference equations as follows:(18)βn,qx,1-βn,qx=nq!xn-1n≥1.
If we putx=0, then βn,q(0,1)-βn,q(0)=δn,1.


## 2.q-Analogue of Euler-Maclaurin Formula

This section introducesq-operator to find q-analogue of Euler-Maclaurin formula. The q-analogue of Euler-Maclaurin formula has been studied in [9]. The authors of [9] appliedq-integral by parts to reachq-analogue of Euler-Maclaurin formula. We can not apply that approach to approximation, because it was written in terms of p(x)=βn,q(x-x). Moreover, the errors have not been studied and our q-operator, which is totally new, leads us to the more applicable function. That is why, this study situates q-Taylor theorem first [8].Theorem 3.
If the functionf(x) is capable of expansion as a convergent power series and if q≠ root of unity, then(19)fx=∑n=0∞x-aqnnq!Dqnfa,where Dq(f(x))=f(xq)-f(x)/x(q-1) is q-derivative of f(x), and for x≠0, we can define it at x=0 as a normal derivative.Definition 4.
One definesHq and Dh operators as follows:(20)Hqnx=h1-qh1-q+xh1-q+2qx⋯h1-q+n-1qx1-qn=x+hnxx+h;nq=∑k=0nnkqqkk-1/2x+hn-k-xkDhFx=Fx+h-Fxh=fx.
In this definition, we assume thatn∈N and the functions and values are well defined h≠0,q≠1. 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. TheH-operator rules like a bridge between the ordinary expansions and q-analogue of these expansions. We may rewrite several formulae of these areas to the form of q-calculus such as [10–14]. Actually we may write q-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” [13].Theorem 5 (fundamental theorem ofh-calculus).
IfF(x) is an h-derivative of f(x) and b-a∈hZ, one has [7](21)∫fxdhx=Fb-Fa,where we define h-integral as follows:(22)∫fxdhx=hfa+fa+h+⋯+fb-hifa<b0ifa=b-hfb+fb+h+⋯+fa-hifa>b.Now in the aid ofq-Taylor expansion, we may write F(x+h) as follows:(23)Fx+h=∑j=0∞x+h-xqjjq!DqjFx=∑j=0∞DqjHqjjq!Fx=eqDqHqFx.Here,eq is q-shifted operator and can be expressed as expansion of Dq and Hq. We may assume that DhF(x)=f(x) and then(24)fx=Fx+h-Fxh=eqDqHq-1HqFx.Since the Jackson integral isq-antiderivative, we have(25)HqDqeqDqHq-1∫fxdqx=Fx.And in the aid of (2), the left hand side of the equation can be expressed as a q-Bernoulli numbers; therefore(26)Fx=∑n=0∞βn,qHqDqnnq!∫fxdqx(27)Fx=∫fxdqx-h2qfx+∑n=1∞βn,qnq!Dqn-1fxHqnx(28)Fx=∫fxdqx-h2qfx+∑n=1∞∑k=0nnkqqkk-1/2h+xn-k-xkDqn-1fxβn,qnq!.Thus we can state the followingq-analogue of Euler-Maclaurin formula.Theorem 6.
If the functionf(x) is capable of expansion as a convergent power series and f(x) decreases so rapidly with x such that all normal derivatives approach zero as x→∞, then one can express the series of function f(x) as follows:(29)∑n=a∞fn=∫a∞fxdqx+12qfa-limb→∞∑n=1∞∑k=0nnkqqkk-1/21+bn-k-bk-1+an-k-akhnDqn-1faβn,qnq!.Proof.
In the aid of Theorem (12) and (24), if we suppose that h=1 and b-a∈N, we have(30)∑n=ab-1fn=∫abfxd1x=Fb-Fa=∫abfxdqx-12qfb-fa+∑n=1∞∑k=0nnkqqkk-1/21+bn-k-bk-1+an-k-akDqn-1fb-Dqn-1faβn,qnq!.
Letf(x) decrease so rapidly with x, since Dqf(x)=(f(xq)-f(x))/x(q-1)), then by applying the limits when x tends to infinity, Dqf(x) reach zero. Now if normal derivatives of f(x) tend to zero, then q-derivatives are also tending to zero. The same discussion makes Dqn-1f(b)=0 for big enough b. Therefore(31)∑n=a∞fn=∫a∞fxdqx+12qfa-limb→∞∑n=1∞∑k=0nnkqqkk-1/21+bn-k-bk-1+an-k-akhnDqn-1faβn,qnq!.Example 7.
Letfx=xs, where s is a positive integer, then we can apply this function at (28), where a=0 and b-1∈N(32)∑n=0b-1ns=bs+1s+1q-bs2q+∑n=2s+1∑k=0nnkqqkk-1/21+bn-k-bksn-1qbs-n+1βn,qnq.
This relation shows the sum of power in the combination ofq-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 [15].Example 8.
Letfx=eq-x=1/Eqx, then this function decreases so rapidly with x such that all normal derivatives approach zero as x→∞. For confirming this, let us mention that Eqx, for some fixed 0<q<1 and -x<1/1-q, is increasing rapidly, since d/dxEqx=∑j=1∞‍1-qqj∏k=0k≠j∞‍1+1-qqkx>0.(33)∑n=0∞eq-n=1+12q-limb→∞∑n=1∞∑k=0nnkqqkk-1/21+bn-k-bkβn,qnq!.
Moreover, this isq-analogue of ∑n=0∞‍e-n=-1/e-1-1=3/2+∑n=1∞‍β2n/2n!, where β2n is a normal Bernoulli number that is generated by -1/e-1-1.


## 3. Approximation byq-Bernoulli Polynomial

We know that the class ofq-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 8.11 [16], 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=Lq20,1=fq:0,1→0,1∣∫01‍fq2(t)dqt<∞ be the Hilbert space [17], then βn,q(x)∈H for n=0,…,N and Y=Spanβ0,q(x),β1,q(x),…,βN,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^2,q≤h-y2,q, where the norm is defined by f2,q≔∫01‍fq2(t)dqt1/2. Following proposition determine the coefficient of q-Bernoulli polynomials, when we estimate any f∈Lq20,1 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 (25), but with the infinite sum on the right hand side repla by the Nth partial sum SN, plus an additional term RN.Supposea∈Z and b=a+1. Consider the Nth partial sum(34)SN=∑k=0Nβk,qkq!HqkDqkfa+1-HqkDqkfa.Letg(x,y)=βN,q(x,y)/Nq!, then in the aid of Lemma 1 we have Dqn-kg(x,y)=βk,q(x,y)/kq! and we have(35)SN=∑k=0Nβk,q0,0kq!HqkDqkfa+1-βk,q0,1kq!HqkDqkfa-Dqfa=∑k=0NDqN-kg0,0HqkDqkfa+1-DqN-kg0,1HqkDqkfa-Dqfa=Dqfa-∑k=0NDqN-kg0,xHqkDqkfa+1-xx=0x=1.By takingq-derivative of the summation, we may rewrite this partial sum by integral presentation. Moreover this relation gives a boundary for RN(x).Proposition 9.
Letf∈Lq20,1 and be estimated by truncated q-Bernoulli series ∑n=0N‍Cnβn,q(x). Then Cn coefficients for n=0,1,…,N can be calculated as an integral form Cn=1/nq!∫01‍Dq(n)f(x)dqx. In addition one can write f(x) in the following form:(36)fx=∫01fxdqx-h2qfx+∑n=1N1nq!∫01Dqnfxdqxβn,qx+Rq,nx.
MoreoverRq,n(x) as a reminder part is bounded by 2n/Nq!supx∈0,1|βn,q(x)|supx∈0,1|Dq(n)f(x)|.Proof.
In fact we approximatef(x) as a linear combination of q-Bernoulli polynomials and we assume that f(x)≃∑n=0N‍Cnβn,q(x), so taking the Jackson integral from both sides leads us to(37)∫01fxdqx≃∑n=0NCn∫01βn,qxdqx=C0∫01β0,qxdqx+C1∫01β1,qxdqx+⋯+CN∫01βN,qxdqx.
In the aid of Lemma1 part (d), all the terms at the right side except the first one have to be zero. We calculate β0,q(x)=1, so C0=1/0q!∫01‍f(x)dqx. Using part (a) of that lemma for q-derivative of q-Bernoulli polynomial gathered by taking q-derivative of f(x) leads to(38)∫01Dqfxdqx≃∑n=0NCnnq!∫01βn-1,qxdqx=C1∫01β0,qxdqx=C1.
Repeating this proceduren-times yields the form of Cn as we mention it at theorem. We mention that if x∈0,1 and 0<|q|<1, then Hqn(x) for h=1 is bounded by 2n. 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 ofq-Bernoulli polynomials. In spite of a lot of investigations on q-Bernoulli polynomials, 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 [3] or [4]. The techniques of H-operator can be applied in operator theory.


## 3.1. Further Works

In this paper, we introduced a proper tool to approximate a given capable function by combinations ofq-Bernoulli polynomials. In spite of a lot of investigations on q-Bernoulli polynomials, 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 [3] or [4]. The techniques of H-operator can be applied in operator theory.



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