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)
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.
The differential equation
was introduced in , 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 ; play the normalizing constant role.
Clearly, the value of distribution vanishes at in each of the above mentioned four cases, i.e., for .
As a special case of (4), let us consider the values , , and 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 , these polynomials are finitely orthogonal with respect to the second kind of beta weight function on if and only if , i.e., we have
if , , where , , and . Moreover, they satisfy a three-term recurrence relation
For example, if , , , and in (10), then the arbitrary function can be approximated as
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
Consider the general form of a weighted quadrature
It can be shown in (15) that for any linear combination of the sequence if and only if are the roots of orthogonal polynomials of degree n with respect to the weight function , and belong to ; see  for more details. Also, it is proved that to derive in (15), when , it is not required to solve the following linear system of order :
Rather, one can directly use the relation
in which is the orthonormal polynomial of , i.e.,
Now, by noting that the symmetric polynomials (10) are finitely orthogonal with respect to the weight function on the real line, we consider the following finite class of quadrature rules:
where are the roots of polynomials and are calculated by
2.1 An important remark
The change of variable in the left-hand side of (20) first changes the interval to such that we have
As the right-hand integral of (22) shows, the shifted Jacobi weight function has appeared for and . Hence, the shifted Gauss-Jacobi quadrature rule [6, 9] with the special parameters and can also be applied for estimating (22). This procedure eventually changes (20) into the form
where are the zeros of shifted Jacobi polynomials on . But, there is the main problem for the formula (23). From (16), it is generally known that the residue of quadrature rules depends on ; . Therefore, by noting (23), we should have
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.
Consider the two-point quadrature formula
in which , , and . According to the explained comments, (25) must be exact for all elements of the basis if and only if , are two roots of . As a particular sample, let us take and . Then (25) is reduced to
in which and are zeros of and , are computed by solving the linear system
After deriving , in (27), the complete form of (26) would be
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 , , and must be satisfied. For instance, if and , then after some computations, the related formula takes the form
and , , are the roots of .
Example 3 To derive a four-point formula of type (20), first the conditions , , and must be satisfied. For example, if and , then
This formula is exact for all elements of the basis ; and its nodes are the roots of .
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.2006.02.007
Masjed-Jamei M: Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation. Integral Transforms Spec. Funct. 2002, 13: 169–190. 10.1080/10652460212898
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.2008.03.030
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.2009.11.010
Masjed-Jamei M, Hussain N: More results on a functional generalization of the Cauchy-Schwarz inequality. J. Inequal. Appl. 2012., 2012: Article ID 239
Davis R, Rabinowitz P: Methods of Numerical Integration. 2nd edition. Academic Press, New York; 1984.
Gautschi W: Construction of Gauss-Christoffel quadrature formulas. Math. Comput. 1968, 22: 251–270. 10.1090/S0025-5718-1968-0228171-0
Krylov VI: Approximate Calculation of Integrals. Macmillan Co., New York; 1962.
Masjed-Jamei M, Kutbi MA, Hussain N: Some new estimates for the error of Simpson integration rule. Abstr. Appl. Anal. 2012., 2012: Article ID 239695
Stoer J, Bulirsch R: Introduction to Numerical Analysis. 2nd edition. Springer, New York; 1993.
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.
The authors declare that they have no competing interests.
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
About this article
Cite this article
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