On orthogonal polynomials and quadrature rules related to the second kind of beta distribution
© Masjed-Jamei and Hussain; licensee Springer. 2013
Received: 2 February 2013
Accepted: 15 March 2013
Published: 5 April 2013
We consider a finite class of weighted quadratures with the weight function on , 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 , , and must be satisfied for its orthogonality relation. Some analytic examples are given in this sense.
MSC:41A55, 65D30, 65D32.
KeywordsGauss-Jacobi quadrature rules weight function second kind of beta distribution dual symmetric distributions family symmetric orthogonal polynomials
where ; play the normalizing constant role.
Clearly, the value of distribution vanishes at in each of the above mentioned four cases, i.e., for .
This means that the finite set is a basis space for all polynomials of degree at most three, i.e., for , 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 in quadrature rules
2.1 An important remark
where are real functions to be computed and , 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 ; . This is the main disadvantage of using (23), which shows the importance of the polynomials (10) in estimating a class of weighted quadrature rules . The following examples clarify this remark.
Relation (28) shows that it is exact for any arbitrary polynomial of degree at most three.
and , , are the roots of .
This formula is exact for all elements of the basis ; and its nodes are the roots of .
Numerical results for two-point formula ( 28 )
Approximate value (2-point)
Numerical results for three-point formula ( 30 )
Approximate value (3-point)
Numerical results for four-point formula ( 32 )
Approximate value (4-point)
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.
- Masjed-Jamei M: A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions. J. Math. Anal. Appl. 2007, 325: 753–775. 10.1016/j.jmaa.2006.02.007MathSciNetView ArticleGoogle Scholar
- Masjed-Jamei M: Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation. Integral Transforms Spec. Funct. 2002, 13: 169–190. 10.1080/10652460212898MathSciNetView ArticleGoogle Scholar
- Masjed-Jamei M, Srivastava HM: An integral expansion for analytic functions based upon the remainder values of the Taylor series expansions. Appl. Math. Lett. 2009, 22: 406–411. 10.1016/j.aml.2008.03.030MathSciNetView ArticleGoogle Scholar
- Masjed-Jamei M, Srivastava HM: Application of a new integral expansion for solving a class of functional equations. Appl. Math. Lett. 2010, 23: 421–425. 10.1016/j.aml.2009.11.010MathSciNetView ArticleGoogle Scholar
- Masjed-Jamei M, Hussain N: More results on a functional generalization of the Cauchy-Schwarz inequality. J. Inequal. Appl. 2012., 2012: Article ID 239Google Scholar
- Davis R, Rabinowitz P: Methods of Numerical Integration. 2nd edition. Academic Press, New York; 1984.Google Scholar
- Gautschi W: Construction of Gauss-Christoffel quadrature formulas. Math. Comput. 1968, 22: 251–270. 10.1090/S0025-5718-1968-0228171-0MathSciNetView ArticleGoogle Scholar
- Krylov VI: Approximate Calculation of Integrals. Macmillan Co., New York; 1962.Google Scholar
- Masjed-Jamei M, Kutbi MA, Hussain N: Some new estimates for the error of Simpson integration rule. Abstr. Appl. Anal. 2012., 2012: Article ID 239695Google Scholar
- Stoer J, Bulirsch R: Introduction to Numerical Analysis. 2nd edition. Springer, New York; 1993.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.