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Special Families of Polynomials 11
Figure 1.5.1 - Chebyshev polynomials Figure 1.5.2 - The eleventh Chebyshev
for 1 ≤ n ≤ 6. polynomial.
Other simple relations can be easily established:
(1.5.9) Tn(x) = 1 i n θ −i n θ
e + e
2
= 1
x +
2
x2 n
− 1 + x − x2 n − 1 ,
(1.5.10) Tn = 1
′ T n+1
2 n + 1 − T ′
n−1
n − 1
(1.5.11) Tm Tn(x) = Tmn (x) , ∀x ∈
, n ≥ 2,
We refer to luke(1969) and rivlin(1974) for more details.
Ī, ∀n, m ∈ N.
x = cos θ ∈ Ī, n ∈ N,
Another family of polynomials, known as Chebyshev polynomials of the second kind
is defined as
(1.5.12) Un := 1
n + 1 T ′ n+1 , n ∈ N.