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On orthogonal polynomials and quadrature rules related to the second kind of beta distribution

Abstract

We consider a finite class of weighted quadratures with the weight function x 2 a ( 1 + x 2 ) b 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 {maxn}a+b1/2, a<1/2, b>0 and ( 1 ) 2 a =1 must be satisfied for its orthogonality relation. Some analytic examples are given in this sense.

MSC:41A55, 65D30, 65D32.

1 Introduction

The differential equation

(1)

was introduced in [1], and it was established that the symmetric polynomials

Φ n ( x ) = S n ( r s p q | x ) = k = 0 [ n / 2 ] ( k ) ( i = 0 [ n / 2 ] ( k + 1 ) ( 2 i + ( 1 ) n + 1 + 2 [ n / 2 ] ) p + r ( 2 i + ( 1 ) n + 1 + 2 ) q + s ) x n 2 k
(2)

are a basis solution of it. If this equation is written in a self-adjoint form, then the first-order equation

x d d x ( ( p x 2 + q ) W ( x ) ) = ( r x 2 + s ) W(x)
(3)

would appear. The solution of equation (3) is known as an analogue of Pearson distributions family and can be indicated as

W ( r s p q | x ) =exp ( ( r 2 p ) x 2 + s x ( p x 2 + q ) d x ) .
(4)

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:

(5)
(6)
(7)
(8)

where K i ; i=1,2,3,4 play the normalizing constant role.

Clearly, the value of distribution vanishes at x=0 in each of the above mentioned four cases, i.e., W(p,q,r,s;0)=0 for s0.

As a special case of (4), let us consider the values p=1, q=1, r=2a2b+2 and s=2a corresponding to distribution (7) and replace them in equation (1) to get

(9)

By solving equation (9), the polynomial solution of monic type is derived

(10)

According to [1], these polynomials are finitely orthogonal with respect to the second kind of beta weight function x 2 a ( 1 + x 2 ) b on (,) if and only if {maxn}a+b1/2, i.e., we have

(11)

if m,n=0,1, , Na+b1/2, where N=max{m,n}, δ n , m ={ 0 ( n m ) , 1 ( n = m ) , a<1/2, b>0 and ( 1 ) 2 a =1. Moreover, they satisfy a three-term recurrence relation

(12)

The orthogonality property (11) shows that the polynomials S ¯ n (1,1,2a2b+2,2a;x) are a suitable tool to finitely approximate the functions that satisfy the Dirichlet conditions [25].

For example, if N={maxn}=3, a+b7/2, a<1/2, b>0 and ( 1 ) 2 a =1 in (10), then the arbitrary function f(x) can be approximated as

f(x) m = 0 3 B m S ¯ m (1,1,2a2b+2,2a;x),
(13)

where

B m = ( ( 1 ) m i = 1 m ( i ( 1 ( 1 ) i ) a ) ( i ( 1 ( 1 ) i ) a 2 b ) ( 2 i 2 a 2 b + 1 ) ( 2 i 2 a 2 b 1 ) ) Γ ( b + a 1 / 2 ) Γ ( a + 1 / 2 ) Γ ( b ) × x 2 a ( 1 + x 2 ) b S ¯ m ( 2 a 2 b + 2 2 a 1 1 | x ) f ( x ) d x .
(14)

This means that the finite set { S ¯ i ( 1 , 1 , 2 a 2 b + 2 , 2 a ; x ) } i = 0 3 is a basis space for all polynomials of degree at most three, i.e., for f(x)= a 3 x 3 + a 2 x 2 + a 1 x+ a 0 , the approximation (13) is exact. This matter helps us to state the application of symmetric orthogonal polynomials (10) in weighted quadrature rules [69].

2 Application of S ¯ n (1,1,2a2b+2,2a;x) in quadrature rules

Consider the general form of a weighted quadrature

α β w(x)f(x)dx= i = 1 n w i f( x i )+ k = 1 m v k f( z k )+ R n , m [f],
(15)

where w(x) is a positive function on [α,β]; { w i } i = 1 n , { v k } k = 1 m are unknown coefficients; { x i } i = 1 n are unknown nodes; { z k } k = 1 m are pre-determined nodes [7, 8]; and finally, the residue R n , m [f] is determined (see, e.g., [8]) by

R n , m [f]= f ( 2 n + m ) ( ξ ) ( 2 n + m ) ! α β w(x) k = 1 m (x z k ) i = 1 n ( x x i ) 2 dx;α<ξ<β.
(16)

It can be shown in (15) that R n , m [f]=0 for any linear combination of the sequence {1,x,, x 2 n + m + 1 } if and only if { x i } i = 1 n are the roots of orthogonal polynomials of degree n with respect to the weight function w(x), and { z k } k = 1 m belong to [α,β]; see [7] for more details. Also, it is proved that to derive { w i } i = 1 n in (15), when m=0, it is not required to solve the following linear system of order n×n:

i = 1 n w i x i j = α β w(x) x j dxfor j=0,1,,2n1.
(17)

Rather, one can directly use the relation

1 w i = P ˆ 0 2 ( x i )+ P ˆ 1 2 ( x i )++ P ˆ n 1 2 ( x i )for i=1,2,,n,
(18)

in which P ˆ i (x) is the orthonormal polynomial of P i (x), i.e.,

P ˆ i (x)= ( α β w ( x ) P i 2 ( x ) d x ) 1 / 2 P i (x).
(19)

Now, by noting that the symmetric polynomials (10) are finitely orthogonal with respect to the weight function W(x,a,b)= x 2 a ( 1 + x 2 ) b on the real line, we consider the following finite class of quadrature rules:

(20)

where x j are the roots of polynomials S ¯ n (1,1,2a2b+2,2a;x) and w j are calculated by

1 w j = i = 0 n 1 ( S ¯ i ( 1 , 1 , 2 a 2 b + 2 , 2 a ; x j ) ) 2 for j=0,1,2,,n.
(21)

2.1 An important remark

The change of variable x= t 1 / 2 ( 1 t ) 1 / 2 in the left-hand side of (20) first changes the interval (,) to [0,1] such that we have

x 2 a ( 1 + x 2 ) b f(x)dx= 0 1 t a + b 3 2 ( 1 t ) a 1 2 f ( 1 t 1 ) dt.
(22)

As the right-hand integral of (22) shows, the shifted Jacobi weight function ( 1 x ) u x v has appeared for u=a1/2 and v=a+b3/2. Hence, the shifted Gauss-Jacobi quadrature rule [6, 9] with the special parameters u=a1/2 and v=a+b3/2 can also be applied for estimating (22). This procedure eventually changes (20) into the form

(23)

where x j ( a 1 / 2 , a + b 3 / 2 ) are the zeros of shifted Jacobi polynomials P n , + ( a 1 / 2 , a + b 3 / 2 ) (x) on [0,1]. But, there is the main problem for the formula (23). From (16), it is generally known that the residue of quadrature rules depends on f ( 2 n ) (ξ); α<ξ<β. Therefore, by noting (23), we should have

d 2 n f ( x 1 1 ) d x 2 n = i = 0 2 n φ i (x) f ( i ) ( x 1 1 ) ,
(24)

where φ i are real functions to be computed and f ( i ) , i=0,1,2,,2n 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 f(x)= x j ; j=0,1,2,,2n1. This is the main disadvantage of using (23), which shows the importance of the polynomials (10) in estimating a class of weighted quadrature rules [10]. The following examples clarify this remark.

Example 1

Consider the two-point quadrature formula

x 2 a ( 1 + x 2 ) b f(x)dx w 1 f( x 1 )+ w 2 f( x 2 ),
(25)

in which a+b5/2, a<1/2, b>0 and ( 1 ) 2 a =1. According to the explained comments, (25) must be exact for all elements of the basis f(x)={1,x, x 2 , x 3 } if and only if x 1 , x 2 are two roots of S ¯ 2 (1,1,2a2b+2,2a;x). As a particular sample, let us take a=0 and b=3. Then (25) is reduced to

1 ( 1 + x 2 ) 3 f(x)dx w 1 f ( 3 3 ) + w 2 f ( 3 3 ) ,
(26)

in which 3 /3 and 3 /3 are zeros of S ¯ 2 (1,1,4,0;x) and w 1 , w 2 are computed by solving the linear system

w 1 + w 2 = ( 1 + x 2 ) 3 dx= 3 8 π, 3 3 ( w 1 w 2 )= x ( 1 + x 2 ) 3 dx=0.
(27)

After deriving w 1 , w 2 in (27), the complete form of (26) would be

1 ( 1 + x 2 ) 3 f(x)dx= 3 π 16 ( f ( 3 3 ) + f ( 3 3 ) ) + R 2 [f],
(28)

where

R 2 [f]= f ( 4 ) ( ξ ) 4 ! 1 ( 1 + x 2 ) 3 ( S ¯ 2 ( 4 0 1 1 | x ) ) 2 dx= π 72 f ( 4 ) (ξ),ξR.
(29)

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 a+b7/2, a<1/2, b>0 and ( 1 ) 2 a =1 must be satisfied. For instance, if a=1 and b=5, then after some computations, the related formula takes the form

x 2 ( 1 + x 2 ) 5 f(x)dx= π 1 , 280 ( 9 f ( 5 3 ) + 32 f ( 0 ) + 9 f ( 5 3 ) ) + R 3 [f],
(30)

where

R 3 [f]= f ( 6 ) ( ξ ) 6 ! x 2 ( 1 + x 2 ) 5 ( S ¯ 3 ( 6 2 1 1 | x ) ) 2 dx= 5 π 3 , 456 f ( 6 ) (ξ),ξR,
(31)

and x 1 = 5 / 3 , x 2 =0, x 3 = 5 / 3 are the roots of S ¯ 3 (1,1,6,2;x)= x 3 (5/3)x.

Example 3 To derive a four-point formula of type (20), first the conditions a+b9/2, a<1/2, b>0 and ( 1 ) 2 a =1 must be satisfied. For example, if a=0 and b=6, then

(32)

where

R 4 [ f ] = f ( 8 ) ( ξ ) 8 ! 1 ( 1 + x 2 ) 6 ( S ¯ 4 ( 10 0 1 1 | x ) ) 2 d x = π 2 , 822 , 400 f ( 8 ) ( ξ ) , ξ R .
(33)

This formula is exact for all elements of the basis f(x)= x j ; j=0,1,2,,7 and its nodes are the roots of S ¯ 4 (1,0,8,2;x)= x 4 (6/5) x 2 +3/35.

Tables 1-3 show some numerical examples related to three given examples.

Table 1 Numerical results for two-point formula ( 28 )
Table 2 Numerical results for three-point formula ( 30 )
Table 3 Numerical results for four-point formula ( 32 )

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Acknowledgements

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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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

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Keywords

  • Gauss-Jacobi quadrature rules
  • weight function
  • second kind of beta distribution
  • dual symmetric distributions family
  • symmetric orthogonal polynomials