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# On orthogonal polynomials and quadrature rules related to the second kind of beta distribution

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 157 (2013)

## Abstract

We consider a finite class of weighted quadratures with the weight function {x}^{-2a}{(1+{x}^{2})}^{-b} on (-\mathrm{\infty},\mathrm{\infty}), which is valid only for finite values of *n* (the number of nodes). This means that classical Gauss-Jacobi quadrature rules cannot be considered for this class, because some restrictions such as \{maxn\}\le a+b-1/2, a<1/2, b>0 and {(-1)}^{2a}=1 must be satisfied for its orthogonality relation. Some analytic examples are given in this sense.

**MSC:**41A55, 65D30, 65D32.

## 1 Introduction

The differential equation

was introduced in [1], and it was established that the symmetric polynomials

are a basis solution of it. If this equation is written in a self-adjoint form, then the first-order equation

would appear. The solution of equation (3) is known as an analogue of Pearson distributions family and can be indicated as

There are four main sub-classes of distributions family (4) (and consequently, sub-solutions of equation (3)) whose explicit probability density functions are, respectively, as follows:

where {K}_{i}; i=1,2,3,4 play the normalizing constant role.

Clearly, the value of distribution vanishes at x=0 in each of the above mentioned four cases, *i.e.*, W(p,q,r,s;0)=0 for s\ne 0.

As a special case of (4), let us consider the values p=1, q=1, r=-2a-2b+2 and s=-2a corresponding to distribution (7) and replace them in equation (1) to get

By solving equation (9), the polynomial solution of monic type is derived

According to [1], these polynomials are finitely orthogonal with respect to the second kind of beta weight function {x}^{-2a}{(1+{x}^{2})}^{-b} on (-\mathrm{\infty},\mathrm{\infty}) if and only if \{maxn\}\le a+b-1/2, *i.e.*, we have

if m,n=0,1,\dots , N\le a+b-1/2, where N=max\{m,n\}, {\delta}_{n,m}=\{\begin{array}{cc}0\hfill & (n\ne m),\hfill \\ 1\hfill & (n=m),\hfill \end{array} a<1/2, b>0 and {(-1)}^{2a}=1. Moreover, they satisfy a three-term recurrence relation

The orthogonality property (11) shows that the polynomials {\overline{S}}_{n}(1,1,-2a-2b+2,-2a;x) are a suitable tool to finitely approximate the functions that satisfy the Dirichlet conditions [2–5].

For example, if N=\{maxn\}=3, a+b\ge 7/2, a<1/2, b>0 and {(-1)}^{2a}=1 in (10), then the arbitrary function f(x) can be approximated as

where

This means that the finite set {\{{\overline{S}}_{i}(1,1,-2a-2b+2,-2a;x)\}}_{i=0}^{3} is a basis space for all polynomials of degree at most three, *i.e.*, for f(x)={a}_{3}{x}^{3}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}, the approximation (13) is exact. This matter helps us to state the application of symmetric orthogonal polynomials (10) in weighted quadrature rules [6–9].

## 2 Application of {\overline{S}}_{n}(1,1,-2a-2b+2,-2a;x) in quadrature rules

Consider the general form of a weighted quadrature

where w(x) is a positive function on [\alpha ,\beta ]; {\{{w}_{i}\}}_{i=1}^{n}, {\{{v}_{k}\}}_{k=1}^{m} are unknown coefficients; {\{{x}_{i}\}}_{i=1}^{n} are unknown nodes; {\{{z}_{k}\}}_{k=1}^{m} are pre-determined nodes [7, 8]; and finally, the residue {R}_{n,m}[f] is determined (see, *e.g.*, [8]) by

It can be shown in (15) that {R}_{n,m}[f]=0 for any linear combination of the sequence \{1,x,\dots ,{x}^{2n+m+1}\} if and only if {\{{x}_{i}\}}_{i=1}^{n} are the roots of orthogonal polynomials of degree *n* with respect to the weight function w(x), and {\{{z}_{k}\}}_{k=1}^{m} belong to [\alpha ,\beta ]; see [7] for more details. Also, it is proved that to derive {\{{w}_{i}\}}_{i=1}^{n} in (15), when m=0, it is not required to solve the following linear system of order n\times n:

Rather, one can directly use the relation

in which {\stackrel{\u02c6}{P}}_{i}(x) is the orthonormal polynomial of {P}_{i}(x), *i.e.*,

Now, by noting that the symmetric polynomials (10) are finitely orthogonal with respect to the weight function W(x,a,b)={x}^{-2a}{(1+{x}^{2})}^{-b} on the real line, we consider the following finite class of quadrature rules:

where {x}_{j} are the roots of polynomials {\overline{S}}_{n}(1,1,-2a-2b+2,-2a;x) and {w}_{j} are calculated by

### 2.1 An important remark

The change of variable x={t}^{-1/2}{(1-t)}^{1/2} in the left-hand side of (20) first changes the interval (-\mathrm{\infty},\mathrm{\infty}) to [0,1] such that we have

As the right-hand integral of (22) shows, the shifted Jacobi weight function {(1-x)}^{u}{x}^{v} has appeared for u=-a-1/2 and v=a+b-3/2. Hence, the shifted Gauss-Jacobi quadrature rule [6, 9] with the special parameters u=-a-1/2 and v=a+b-3/2 can also be applied for estimating (22). This procedure eventually changes (20) into the form

where {x}_{j}^{(-a-1/2,a+b-3/2)} are the zeros of shifted Jacobi polynomials {P}_{n,+}^{(-a-1/2,a+b-3/2)}(x) on [0,1]. But, there is the main problem for the formula (23). From (16), it is generally known that the residue of quadrature rules depends on {f}^{(2n)}(\xi ); \alpha <\xi <\beta. Therefore, by noting (23), we should have

where {\phi}_{i} are real functions to be computed and {f}^{(i)}, i=0,1,2,\dots ,2n are the successive derivatives of the function *f*. On the other hand, the function *f* cannot be in the form of an arbitrary polynomial in order that the right-hand side of (24) becomes zero. In other words, the formula (23) cannot be exact for all elements of the basis f(x)={x}^{j}; j=0,1,2,\dots ,2n-1. This is the main disadvantage of using (23), which shows the importance of the polynomials (10) in estimating a class of weighted quadrature rules [10]. The following examples clarify this remark.

**Example 1**

Consider the two-point quadrature formula

in which a+b\ge 5/2, a<1/2, b>0 and {(-1)}^{2a}=1. According to the explained comments, (25) must be exact for all elements of the basis f(x)=\{1,x,{x}^{2},{x}^{3}\} if and only if {x}_{1}, {x}_{2} are two roots of {\overline{S}}_{2}(1,1,-2a-2b+2,-2a;x). As a particular sample, let us take a=0 and b=3. Then (25) is reduced to

in which \sqrt{3}/3 and -\sqrt{3}/3 are zeros of {\overline{S}}_{2}(1,1,-4,0;x) and {w}_{1}, {w}_{2} are computed by solving the linear system

After deriving {w}_{1}, {w}_{2} in (27), the complete form of (26) would be

where

Relation (28) shows that it is exact for any arbitrary polynomial of degree at most three.

**Example 2** To have a three-point formula of type (20), first we should note that the conditions a+b\ge 7/2, a<1/2, b>0 and {(-1)}^{2a}=1 must be satisfied. For instance, if a=-1 and b=5, then after some computations, the related formula takes the form

where

and {x}_{1}=\sqrt{5/3}, {x}_{2}=0, {x}_{3}=-\sqrt{5/3} are the roots of {\overline{S}}_{3}(1,1,-6,2;x)={x}^{3}-(5/3)x.

**Example 3** To derive a four-point formula of type (20), first the conditions a+b\ge 9/2, a<1/2, b>0 and {(-1)}^{2a}=1 must be satisfied. For example, if a=0 and b=6, then

where

This formula is exact for all elements of the basis f(x)={x}^{j}; j=0,1,2,\dots ,7 and its nodes are the roots of {\overline{S}}_{4}(1,0,-8,2;x)={x}^{4}-(6/5){x}^{2}+3/35.

Tables 1-3 show some numerical examples related to three given examples.

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

The work of the first author is supported by a grant from Iran National Science Foundation and the work of the second author has been supported by the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) of Jeddah.

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Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

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Masjed-Jamei, M., Hussain, N. On orthogonal polynomials and quadrature rules related to the second kind of beta distribution.
*J Inequal Appl* **2013**, 157 (2013). https://doi.org/10.1186/1029-242X-2013-157

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DOI: https://doi.org/10.1186/1029-242X-2013-157