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.
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.
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