We introduce a finite class of weighted quadrature rules with the weight function on as , where are the zeros of polynomials orthogonal with respect to the introduced weight function, are the corresponding coefficients, and is the error value. We show that the above formula is valid only for the finite values of . In other words, the condition must always be satisfied in order that one can apply the above quadrature rule. In this sense, some numerical and analytic examples are also given and compared.

1. Introduction

Recently in [1] the differential equation is introduced, and its explicit solution is shown by It is also called the generic equation of classical symmetric orthogonal polynomials [1, 2]. If this equation is written in a self-adjoint form then the first-order equation is derived. The solution of (1.3) is known as an analogue of Pearson distributions family and can be indicated as In general, there are four main subclasses of distributions family (1.4) (as subsolutions of (1.3) whose explicit probability density functions are, respectively, The values play the normalizing constant role in these distributions. Moreover, the value of distribution vanishes at in each four cases, that is, for . Hence, (1.4) is called in [1] β€œThe dual symmetric distributions family.”

As a special case of , let us choose the values and corresponding to distribution (1.8) here and replace them in (1.1) to get If (1.9) is solved, the polynomial solution of monic type

is obtained. According to [1], these polynomials are finitely orthogonal with respect to a special kind of Freud weight function, that is, , on the real line if and only if ; see also [3, 4]. In other words, we have if and only if and Furthermore, the polynomials (1.10) also satisfy a three-term recurrence relation as

But the polynomials are suitable tool to finitely approximate arbitrary functions, which satisfy the Dirichlet conditions (see, e.g., [5]). For example, suppose that and in (1.10). Then, the function can finitely be approximated as


for .

Clearly (1.14) is valid only when the general function in (1.15) is integrable for any . This means that the finite set is a basis space for all polynomials of degree at most three. So if , the approximation (1.14) is exact. By noting this, here is a good position to express an application of the mentioned polynomials in weighted quadrature rules [6, 7] by a straightforward example. Let us consider a two-point approximation as

provided that . According to the described themes, (1.16) must be exact for all elements of the basis if and only if are two roots of . For instance, if then (1.16) should be changed to

in which and are zeros of , and are computed by solving the linear system

Hence, after solving (1.18) the final form of (1.16) is known as

This approximation is exact for all arbitrary polynomials of degree at most 3.

2. Application of Polynomials (1.10) in Weighted Quadrature Rules: General Case

As we know, the general form of weighted quadrature rules is given by

in which the weights and the nodes are unknown values, is a positive function, and is an arbitrary interval; see, for example, [6, 7]. Moreover the residue is determined (see, e.g., [7]) by

It can be proved in (2.1) that for any linear combination of the sequence if and only if are the roots of orthogonal polynomials of degree with respect to the weight function on the interval . For more details, see [6]. Also, it is proved that to derive in (2.1), it is not required to solve the following linear system of order :

rather, one can directly use the relation

where are orthonormal polynomials of defined as

In this way, as it is shown in [8, 9], satisfies a particular type of three-term recurrence as

Now, by noting these comments and the fact that the symmetric polynomials are finitely orthogonal with respect to the weight function on the real line, we can define a finite class of quadrature rules as

in which are the roots of and are computed by

Moreover, for the residue value we have

2.1. An Important Remark

It is important to note that by applying the change of variable in the left-hand side of (2.7) the orthogonality interval changes to and subsequently

As it is observed, the right-hand integral of (2.10) contains the well-known Laguerre weight function for . Hence, one can use Gauss-Laguerre quadrature rules [8, 9] with the special parameter . This process changes (2.7) in the form

in which are the zeros of Laguerre polynomials . But, there is a large disadvantage for formula (2.11). According to (2.2) or (2.9), the residue of integration rules generally depends on . Thus, by noting (2.11) we should have

where are real functions to be computed and are the successive derivatives of function .

As we observe in (2.12), cannot be in the form of an arbitrary polynomial function in order that the right-hand side of (2.12) is equal to zero. In other words, (2.11) is not exact for the basis space . This is the main disadvantage of using (2.11), as the examples of next section support this claim.

3. Examples

Example 3.1. Since a 2-point formula was presented in (1.19), in this example we consider a 3-point integration formula. For this purpose, we should first note that according to (1.11) the condition is necessary. Hence, let us, for instance, assume that . After some computations the related quadrature rule would take the form where and , , and are the roots of . Moreover, can be computed by in which

Example 3.2. To have a 4-point formula, we should again note that is a necessary condition. In this sense, if, for example, then we eventually get where Clearly this formula is exact for the basis elements and the nodes of quadrature (3.5) are the roots of .

4. Numerical results

In this section, some numerical examples are given and compared. The numerical results related to the 2-point formula (1.19) are presented in Table 1, the results related to 3-point formula (3.1) are given in Table 2, and finally the results related to 4-point formula (3.5) are presented in Table 3.