Untitled - Cdm.unimo.it

Numerical Integration 37
In the case of ultraspherical polynomials (− 1
2
refined as follows:
(3.1.2) −1 < −cos
(3.1.3) 0 ≤ cos
k + α
2
− 1
4
n + α + 1
2
π ≤ ξ (n)
k
n − k + 1
n + α + 1 π ≤ ξ
2
(n)
k
Of course, if n is odd we have ξ (n)
(n+1)/2 = 0.
≤ −cos
1
≤ α = β ≤ 2 ), the inequalities can be
k
n + α + 1
2
≤ cos n − k + α
2
n + α + 1
2
π ≤ 0, 1 ≤ k ≤
n
2 ,
+ 3
4
π < 1,
n + 1 −
n
2 ≤ k ≤ n.
Particularly interesting are the following cases, in which an exact expression of the
zeroes is known:
(3.1.4) ξ (n)
k
(3.1.5) ξ (n)
k
(3.1.6) ξ (n)
k
(3.1.7) ξ (n)
k
where 1 ≤ k ≤ n.
2k − 1
= −cos π if α = β = −1
2n 2 ,
k
1
= −cos π if α = β =
n + 1 2 ,
= −cos
2k
2n + 1
1
π if α = , β = −1
2 2 ,
2k − 1
1
= −cos π if α = −1 , β =
2n + 1 2 2 ,
Zeroes in the Legendre case - Estimates indicating the location of the zeroes are obtained
by setting α = 0 in (3.1.2) and (3.1.3). Figure 3.1.1 shows their distribution in the
interval I =] − 1,1[ for n = 7 and n = 10.