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Special Families of Polynomials 9
(1.4.8) |Pn(x)| ≤ 1
π
≤ 1
π
0
π
0
π
x 2 + (1 − x 2 ) cos 2 θ n/2 dθ
2 2 2 1 + x
x + (1 − x ) cos θ dθ = 2
, x ∈ Ī, n ≥ 2.
2
This shows that the Legendre polynomials in
parabolas y = − 1
2 (1 + x2 ) and y = 1
2 (1 + x2 ).
1.5 Chebyshev polynomials
Ī are uniformly bounded between the
The Chebyshev polynomials (of the first kind) are related to the ultraspherical polyno-
mials with α = β = −1 2 . In fact, they are defined by
(− 1
,− 1
2 )
2 (1.5.1) Tn := δn P n , n ∈ N,
where
δn := (n!2n ) 2
(2n)! = n! √ π
Γ(n + 1 =
)
2
1−1
n − 2 .
n
Consequently, they are solutions of the Sturm-Liouville problem
(1.5.2) (1 − x 2 ) T ′′
n − xT ′ n + n 2 Tn = 0 , n ∈ N.
By taking α = β = − 1
2
formula
in (1.3.8), after appropriate scaling, we arrive at the recursion
(1.5.3) Tn(x) = 2xTn−1(x) − Tn−2(x) , x ∈
where T0(x) = 1 and T1(x) = x.
Ī , n ≥ 2,