Weighted quadrature rules with weight function x-pe-1x on [0, ∞)

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Abstract

In this paper, we introduce an analytical class of weighted quadrature rules whose weight function is as W(x,p)=x-pe-1x on [0, ∞).

The formulas given in this work generally obey the following form:0x-pe-1xf(x)dx=i=1nwif(xi)+Rn[f],in which xi are the zeros of polynomials orthogonal with respect to the introduced weight function, wi are the corresponding coefficients and Rn[f] is the related error. It should be pointed out that the above formula is valid only for the finite values of n. In other words, the condition p > 2{max n} + 1 is essential in order to be applicable the above quadrature rule. Some analytical examples are finally given and compared.

Introduction

Let us start our discussion with the following differential equation:(ax2+bx+c)yn(x)+(dx+e)yn(x)-n((n-1)a+d)yn(x)=0;nZ+,which is known as the generic equation of classical orthogonal polynomials of degree n. If this equation is written in self-adjoint form, then the first order equation:dWdx=(d-2a)x+(e-b)ax2+bx+cWis derived. The solution of (2) is known as the Pearson distributions family [4] and can be indicated asWd,ea,b,cx=expdx+eax2+bx+cdx,where d = d  2a and e = e  b.

As a special case, let us consider the values a = 1, b = 0, c = 0, d = 2  p and e = 1. Replacing these values in (1) gives the equation:x2yn(x)+((2-p)x+1)yn(x)-n(n+1-p)yn(x)=0.If (4) is solved the polynomial solutionNn(p)(x)=(-1)nk=0nk!p-(n+1)knn-k(-x)knk=n!k!(n-k)!is derived. According to [7], these polynomials are orthogonal with respect to the weight function W(x,p)=x-pe-1x on [0, ∞) if and only if:p>2{Maxn}+1.In other words, we have:0x-pe-1/xNn(p)(x)Nm(p)(x)dx=n!(p-(n+1))!p-(2n+1)δn,m,if and only if: m, n = 0, 1, 2,  , N < (p  1)/2, δn,m=0ifnm1ifn=m. Moreover, they satisfy the three term recurrence relation:Nn+1(p)(x)=(p-(2n+2))(p-(2n+1))p-(n+1)x-p(p-(2n+1))(p-(n+1))(p-2n)Nn(p)(x)-n(p-(2n+2))(p-(n+1))(p-2n)Nn-1(p)(x),where: N0(p)(x)=1 and N1(p)(x)=(p-2)x-1.

Here we mention that the polynomials Nn(p)(x) are suitable tool to use in the functions approximation theory, provided that those functions satisfy the Dirichlet conditions. Let us consider an example to clarify our purpose.

Suppose n = 3 and p > 7 in the orthogonal polynomials Nn(p)(x). The arbitrary function f(x) can be approximated byf(x)C0N0(p)(x)+C1N1(p)(x)+C2N2(p)(x)+C3N3(p)(x),x[0,),in whichCi=p-(2i+1)i!(p-(i+1))!0x-pe-1xNi(p)(x)f(x)dx,i=0,1,2,3.One observes that the finite set {Ni(p)(x)}i=0i=3, is a basis space for all polynomials of degree at most three. In other words, if f(x) = a3x3 + a2x2 + a1x + a0 is assumed, then the approximation (9) will be exact.

By noting this comment, we can now propound an application of polynomials Nn(p)(x) in the numerical integration theory by a straightforward example. We would like to find an analytical two-point formula as follows:0x-pe-1xf(x)dxw1f(x1)+w2f(x2),provided that p > 5. According to explained themes, (11) must be exact for the basis f(x) = xj; j = 0, 1, 2, 3 if and only if x1, x2 are two roots of N2(p)(x). For instance, if p=152>5, then we get0x-152e-1xf(x)dxw1f6-2221+w2f6+2221,in which 6-2221 and 6+2221 are zeros of N2(152)(x) and w1, w2 are obtained from the linear system:w1+w2=1039564π,6-221w1+6+2221w2=94532π.Solving (13) eventually gives the two-point formula as follows:0x-152e-1xf(x)dx10395π+56702π128f6-2221+10395π-56702π128f6+2221,which is exact for any arbitrary polynomial of degree at most three.

Section snippets

Application of introduced orthogonal polynomials in quadrature rules in general case

As we know the main form of Gauss quadrature rules is given by:abf(x)dw(x)=j=1nwjf(xj)+k=1mvkf(zk)+Rn,m[f],where the weights [wj]j=1n, [vk]k=1m and nodes [xj]j=1n are unknowns and the nodes [zk]k=1m are predetermined. w is [1], [3], [5] also a positive measure on [a, b].

The residue Rn,m[f] is determined (see for instance [8]) by:Rn,m[f]=f(2n+m)(η)(2n+m)!abk=1m(x-zk)j=1n(x-xj)2dw(x),a<η<b.It can be shown that Rn,m[f] = 0, for any linear combination of the sequence {1, x,  , x2n+m−1} if and only

Examples

Example 1

Since we presented a 2-point formula in relation (14), it is better now to propound a 3-point integration formula. For this purpose, according to the main relation (7), the condition p > 7 must be satisfied in order that one can use a 3-point formula. By noting this fact, suppose p = 8. Hence we have:0x-8e-1xf(x)dx=(565.2150607824)f(0.1288864005)+(154.3624193336)f(0.3025345782)+(0.4225198843)f(1.0685790213)+R3[f],whereR3[f]=f(6)(ξ)(24)2×6!0x-8e-1x(N3(8)(x))2dx=f(6)(ξ)2880,0ξ<,and x1 = 

Numerical results

In this section we consider some numerical examples. First, the results obtained by the 2-point formula (14) are presented in Table 1. Then, we have the resutls using the 3-point formula for Example 1 in Table 2. And finally, the results obtained by the 4-point formula for Example 2 are given in Table 3.

Conclusion

In this paper, an approach was introduced to estimate a finite class of weighted quadrature rules with the weight function x-pe-1x on [0,∞). The advantages of presented formulas were also discussed. Some analytical examples were finally given and compared.

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