We introduce a finite class of weighted quadrature rules with the
weight function |x|−2aexp(−1/x2) on (−∞,∞) as
∫−∞∞|x|−2aexp(−1/x2)f(x)dx=∑i=1nwif(xi)+Rn[f], where xi are the zeros of polynomials orthogonal with respect to the introduced weight
function, wi are the corresponding coefficients, and Rn[f] is the error value. We show that the
above formula is valid only for the finite values of n. In other words, the condition a≥{maxn}+1/2 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
x2(px2+q)Φn′′(x)+x(rx2+s)Φn′(x)-(n(r+(n-1)p)x2+(1-(-1)n)s2)Φn(x)=0
is introduced, and its explicit solution is shown by
Sn(rspq|x)=∑k=0[n/2]([n2]k)(∏i=0[n/2]-(k+1)(2i+(-1)n+1+2[n/2])p+r(2i+(-1)n+1+2)q+s)xn-2k.
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
xddx((px2+q)W(x))=(rx2+s)W(x)
is derived. The solution of (1.3) is known as an analogue of Pearson distributions family and can be indicated as
W(rspq|x)=exp(∫(r-2p)x2+sx(px2+q)dx).
In general, there are four main subclasses of distributions family (1.4) (as subsolutions of (1.3) whose explicit probability density functions are, respectively,
K1W(-2a-2b-2,2a-1,1|x)=Γ(a+b+3/2)Γ(a+1/2)Γ(b+1)x2a(1-x2)b,-1≤x≤1,a+12>0,b+1>0,K2W(-2,2a0,1|x)=1Γ(a+1/2)x2aexp(-x2),-∞<x<∞,a+12>0,K3W(-2a-2b+2,-2a1,1|x)=Γ(b)Γ(b+a-1/2)Γ(-a+1/2)x-2a(1+x2)b,-∞<x<∞,b>0,a<12,b+a>12,K4W(-2a+2,21,0|x)=1Γ(a-1/2)x-2aexp(-1x2),-∞<x<∞,a>12.
The values Ki;i=1,2,3,4 play the normalizing constant role in these distributions. Moreover, the value of distribution vanishes at x=0 in each four cases, that is, W(0;p,q,r,s)=0 for s≠0. Hence, (1.4) is called in [1] “The dual symmetric distributions family.”

As a special case of W(x;p,q,r,s), let us choose the values p=1,q=0,r=-2a+2, and s=2 corresponding to distribution (1.8) here and replace them in (1.1) to get
x4Φn′′(x)+2x((1-a)x2+1)Φn′(x)-(n(n+1-2a)x2+1-(-1)n)Φn(x)=0.
If (1.9) is solved, the polynomial solution of monic type

S̅n(-2a+2210|x)=∏i=0[n/2]-122i+2[n/2]+(-1)n+1+2-2a×∑k=0[n/2]([n2]k)(∏i=0[n/2]-(k+1)2i+2[n/2]+(-1)n+1+2-2a2)xn-2k
is obtained. According to [1], these polynomials are finitely orthogonal with respect to a special kind of Freud weight function, that is, x-2aexp(-1/x2), on the real line (-∞,∞) if and only if a≥{maxn}+1/2; see also [3, 4]. In other words, we have
∫-∞∞|x|-2aexp(-1x2)S̅n(-2a+2210|x)S̅m(-2a+2210|x)dx=(∏i=1n2(-1)i(i-a)+2a(2i-2a+1)(2i-2a-1))Γ(a-12)δn,m,
if and only if m,n=0,1,2,…,N=max{m,n}≤a-1/2,(-1)2a=1 and
δn,m={0,ifn≠m,1,ifn=m.
Furthermore, the polynomials (1.10) also satisfy a three-term recurrence relation as

S̅n+1(x)=xS̅n(x)-2(-1)n(n-a)+2a(2n-2a+1)(2n-2a-1)S̅n-1(x),S̅0(x)=1,S̅1(x)=x,n∈ℕ.
But the polynomials S̅n(x;1,0,-2a+2,2) are suitable tool to finitely approximate arbitrary functions, which satisfy the Dirichlet conditions (see, e.g., [5]). For example, suppose that N=max{m,n}=3 and a>7/2 in (1.10). Then, the function f(x) can finitely be approximated as

f(x)≅C0S̅0(x;1,0,-2a+2,2)+C1S̅1(x;1,0,-2a+2,2)+C2S̅2(x;1,0,-2a+2,2)+C3S̅3(x;1,0,-2a+2,2),
where

Cm=∫-∞∞|x|-2aexp(-1/x2)S̅m(-2a+2210|x)f(x)dx(∏i=1m((2(-1)i(i-a)+2a)/(2i-2a+1)(2i-2a-1))Γ(a-1/2)),
for m=0,1,2,3.

Clearly (1.14) is valid only when the general function xm|x|-2aexp(-1/x2)f(x) in (1.15) is integrable for any m=0,1,2,3. This means that the finite set {S̅i(x;1,0,-2a+2,2)}i=03 is a basis space for all polynomials of degree at most three. So if f(x)=a3x3+a2x2+a1x+a0, 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

∫-∞∞|x|-2aexp(-1x2)f(x)dx≅w1f(x1)+w2f(x2),
provided that a>5/2. According to the described themes, (1.16) must be exact for all elements of the basis f(x)={x3,x2,x,1} if and only if x1,x2 are two roots of S̅2(x;1,0,-2a+2,2). For instance, if a=3>5/2 then (1.16) should be changed to

∫-∞∞x-6exp(-1x2)f(x)dx≅w1f(23)+w2f(-23),
in which 2/3 and -2/3 are zeros of S̅2(x;1,0,-4,2), and w1,w2 are computed by solving the linear system

w1+w2=∫-∞∞x-6exp(-1x2)dx=34π,23(w1-w2)=∫-∞∞x-5exp(-1x2)dx=0.
Hence, after solving (1.18) the final form of (1.16) is known as

∫-∞∞x-6exp(-1x2)f(x)dx≅38π(f(23)+f(-23)).
This approximation is exact for all arbitrary polynomials of degree at most 3.

2. Application of Polynomials (<xref ref-type="disp-formula" rid="EEq8">1.10</xref>) in Weighted Quadrature Rules: General Case

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

∫αβw(x)f(x)dx=∑i=1nwif(xi)+Rn[f],
in which the weights {wi}i=1n and the nodes {xi}i=1n are unknown values, w(x) is a positive function, and [α,β] is an arbitrary interval; see, for example, [6, 7]. Moreover the residue Rn[f] is determined (see, e.g., [7]) by

Rn[f]=f(2n)(ξ)(2n)!∫αβw(x)∏i=1n(x-xi)2dx,α<ξ<β.
It can be proved in (2.1) that Rn[f]=0 for any linear combination of the sequence {1,x,x2,…,x2n-1} if and only if {xi}i=1n are the roots of orthogonal polynomials of degree n with respect to the weight function w(x) on the interval [α,β]. For more details, see [6]. Also, it is proved that to derive {wi}i=1n in (2.1), it is not required to solve the following linear system of order n×n:

∑i=1nwixij=∫αβw(x)xjdxforj=0,1,…,2n-1,
rather, one can directly use the relation

1wi=P̂02(xi)+P̂12(xi)+⋯+P̂n-12(xi)fori=1,2,…,n,
where P̂i(x) are orthonormal polynomials of Pi(x) defined as

P̂i(x)=(∫αβw(x)Pi2(x)dx)-1/2Pi(x).
In this way, as it is shown in [8, 9], P̂i(x) satisfies a particular type of three-term recurrence as

xP̂n-1(x)=αnP̂n(x)+βnP̂n-1(x)+αn-1P̂n-2(x).
Now, by noting these comments and the fact that the symmetric polynomials S̅n(x;1,0,-2a+2,2) are finitely orthogonal with respect to the weight function W(x,a)=|x|-2aexp(-1/x2) on the real line, we can define a finite class of quadrature rules as

∫-∞∞|x|-2aexp(-1x2)f(x)dx=∑j=1nwjf(xj)+Rn[f],
in which xj are the roots of S̅n(x;1,0,-2a+2,2) and wj are computed by

1wj=∑i=0n-1(S̅i*(1,0,-2a+2,2;xj))2,forj=0,1,2,…,n.
Moreover, for the residue value we have

It is important to note that by applying the change of variable 1/x2=t in the left-hand side of (2.7) the orthogonality interval (-∞,∞) changes to [0,∞) and subsequently

∫-∞∞|x|-2aexp(-1x2)f(x)dx=∫0∞ta-3/2e-tf(1t)dt.
As it is observed, the right-hand integral of (2.10) contains the well-known Laguerre weight function xue-x for u=a-3/2. Hence, one can use Gauss-Laguerre quadrature rules [8, 9] with the special parameter u=a-3/2. This process changes (2.7) in the form

∫-∞∞|x|-2aexp(-1x2)f(x)dx=∑j=1nwj(a-3/2)f(1xj(a-3/2))+Rn[f(1x)],
in which xj(a-3/2) are the zeros of Laguerre polynomials Ln(a-3/2)(x). 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 f(2n)(ξ);α<ξ<β. Thus, by noting (2.11) we should have

d2nf(1/x)dx2n=∑i=02nϕi(x)f(i)(1x),
where ϕi(x) are real functions to be computed and f(i),i=0,1,2,…,2n, are the successive derivatives of function f(x).

As we observe in (2.12), f(x) 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 f(x)=xj,j=0,1,2,…,2n-1. This is the main disadvantage of using (2.11), as the examples of next section support this claim.

3. ExamplesExample 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 a>7/2 is necessary. Hence, let us, for instance, assume that a=4. After some computations the related quadrature rule would take the form
∫-∞∞x-8exp(-1x2)f(x)dx=316π(3f(23)+4f(0)+3f(-23))+R3[f],
where
R3[f]=f(6)(ξ)6!∫-∞∞x-8exp(-1x2)(S̅3(-6210|x))2dx=π1080f(6)(ξ),ξ∈R,
and x1=2/3, x2=0, and x3=-2/3 are the roots of S̅3(x;1,0,-6,2)=x3-(2/3)x. Moreover, w1,w2,w3 can be computed by
1wj=∑i=02(S̅i*(xj;1,0,-6,2))2,j=1,2,3,
in which
S̅i*(xj;1,0,-6,2)=S̅i(xj;1,0,-6,2)〈S̅i(xj;1,0,-6,2),S̅i(xj;1,0,-6,2)〉1/2.

Example 3.2.

To have a 4-point formula, we should again note that a>9/2 is a necessary condition. In this sense, if, for example, a=5 then we eventually get
∫-∞∞x-10exp(-1x2)f(x)dx=1564π(7-210)(f(10+21015)+f(-10+21015))+1564π(7+210)(f(10-21015)+f(-10-21015))+R4[f],
where
R4[f]=f(8)(ξ)8!∫-∞∞x-10exp(-1x2)(S̅4(-8210|x))2dx=π75600f(8)(ξ),ξ∈R.
Clearly this formula is exact for the basis elements f(x)=xj,j=0,1,2,…,7, and the nodes of quadrature (3.5) are the roots of S̅4(x;1,0,-8,2)=x4-(4/3)x2+4/15.

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.

∫-∞+∞x-6exp(-1/x2)f(x)dx.

f(x)

Approx. value (2-point)

Exact value

Error

cosx

0.9103037512

0.9382539141

0.0279501629

exp(-2/x2)

0.0661839608

0.0852772257

0.0190932649

exp(-cosx)

0.6702559297

0.6812645398

0.0110086101

∫-∞+∞x-8exp(-1/x2)f(x)dx.

f(x)

Approx. value (3-point)

Exact value

Error

exp(-cosx)

1.494420894

1.492841821

0.001579073

1+sinx2

3.866024228

3.866700560

0.000676332

1+cosx2

4.544708979

4.561266761

0.016557782

∫-∞+∞x-10e(-1/x2)f(x)dx.

f(x)

Approx. value (4-point)

Exact value

Error

1+cosx2

16.21776936

16.21978539

0.002016030

(1+x2)-1/2

10.30987753

10.31704740

0.007116987

exp(-x2-2)

1.198219038

1.199125136

0.000906098

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