Untitled - Cdm.unimo.it

Numerical Integration 41
We give the plots of the zeroes of the derivative of Legendre and Chebyshev poly-
nomials for n = 10, in figure 3.1.5.
Figure 3.1.5 - The points η (10)
k , 0 ≤ k ≤ 10, in the cases of Legendre and Chebyshev.
By virtue of (1.5.6) and (1.5.7), Chebyshev polynomials satisfy for n ≥ 1
(3.1.15) Tn(η (n)
j ) = (−1) n+j , 0 ≤ j ≤ n,
(3.1.16) T ′ n(ξ (n)
j ) =
n (−1)n+j
1 − [ξ (n)
j ] 2
, 1 ≤ j ≤ n.
Other useful relations are (n ≥ 1, α ≥ −1, β ≥ −1)
(3.1.17)
(3.1.18)
(3.1.19)
d
dx
d
dx
(α,β)
P n (1) =
P (α,β)
n
(n + α + β + 1) Γ(n + α + 1)
,
2 (n − 1)! Γ(α + 2)
(−1) = − (−1)n (n + α + β + 1) Γ(n + β + 1)
,
2 (n − 1)! Γ(β + 2)
d
dx L(α)
n (0) = −
Γ(n + α + 1)
(n − 1)! Γ(α + 2) .
These equalities are easily obtained from (1.3.1) and (1.6.1).