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

- Mohammad Masjed-Jamei
^{1}and - Nawab Hussain
^{2}Email author

**2013**:157

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

© Masjed-Jamei and Hussain; licensee Springer. 2013

**Received: **2 February 2013

**Accepted: **15 March 2013

**Published: **5 April 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.

### Keywords

Gauss-Jacobi quadrature rules weight function second kind of beta distribution dual symmetric distributions family symmetric orthogonal polynomials## 1 Introduction

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 $s\ne 0$.

*i.e.*, we have

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].

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

*e.g.*, [8]) by

*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 $n\times n$:

*i.e.*,

### 2.1 An important remark

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**

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

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

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 ${\overline{S}}_{4}(1,0,-8,2;x)={x}^{4}-(6/5){x}^{2}+3/35$.

**Numerical results for two-point formula (**
**28**
**)**

$\mathit{f}\mathbf{(}\mathit{x}\mathbf{)}$ | Approximate value (2-point) | Exact value | Error |
---|---|---|---|

$cos{x}^{2}$ | 1.113251175 | 1.041656130 | 0.071595045 |

$exp(-{x}^{2}/2)$ | 0.997237788 | 1.037543288 | 0.040305500 |

exp(−cos | 0.509660126 | 0.519034734 | 0.009374608 |

$\sqrt{1+{x}^{2}}$ | 1.360349524 | 1.333333333 | 0.027016191 |

**Numerical results for three-point formula (**
**30**
**)**

$\mathit{f}\mathbf{(}\mathit{x}\mathbf{)}$ | Approximate value (3-point) | Exact value | Error |
---|---|---|---|

$cos{x}^{2}$ | 0.0743108795 | 0.09326578594 | 0.01895490641 |

$exp(-{x}^{2}/2)$ | 0.09773977703 | 0.09545329274 | 0.00228738430 |

exp(−cos | 0.06241097330 | 0.06149960816 | 0.00091136514 |

$\sqrt{1+{x}^{2}}$ | 0.15068324430 | 0.15238095240 | 0.00169770810 |

**Numerical results for four-point formula (**
**32**
**)**

$\mathit{f}\mathbf{(}\mathit{x}\mathbf{)}$ | Approximate value (4-point) | Exact value | Error |
---|---|---|---|

$cos{x}^{2}$ | 0.7563575358 | 0.7567616833 | 0.0004041475 |

$exp(-{x}^{2}/2)$ | 0.7341056789 | 0.7341611797 | 0.0000555010 |

exp(−cos | 0.3013485879 | 0.3013339743 | 0.0000146136 |

$\sqrt{1+{x}^{2}}$ | 0.8128655892 | 0.8126984127 | 0.0001671765 |

## Declarations

### 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.

## Authors’ Affiliations

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