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18 Chebyshev Polynomials
18.1 Basic Properties and Formulae
The following equations can be found, e.g., in [1, 2].
18.1.1 Definition and Recurrence Formula
The Chebyshev polynomial of first kind T n (x) is a polynomial in x of degree n,
defined by
T n (x) = cos(nϑ) , with x = cos(ϑ) and ϑ ∈ [π, 0].
(18.1a)
Similar the Chebyshev polynomial of second kind U n (x) is defined by
U n (x) =
sin((n + 1)ϑ)
sin(ϑ)
, with x = cos(ϑ) and ϑ ∈ [π, 0],
(18.1b)
the Chebyshev polynomial of third kind V n (x) is defined by
V n (x) =
cos((n +
1
2 )ϑ)
cos(
1
2 ϑ)
, with x = cos(ϑ) and ϑ ∈ [π, 0],
(18.1c)
and finally the Chebyshev polynomial of fourth kind W n (x) is defined via
W n (x) =
sin((n +
1
2 )ϑ)
sin(
1
2 ϑ)
, with x = cos(ϑ) and ϑ ∈ [π, 0].
(18.1d)
Figure 18.1 shows as an example the Chebyshev polynomials up to degree 3.
It is straightforward to derive from the trigonometric identities for multiply
angels the recurrence equation
C n (x) = 2x C n−1 (x) − C n−2 (x) , n = 2, 3, · · · , and
(18.2a)
C n are the Chebyshev polynomials of degree n, thus C n (x) = T n (x) or = U n (x) or
= V n (x) or = W n (x), and the initial polynomial values are
T 0 (x) = 1 , T 1 (x) = x ; U 0 (x) = 1 , U 1 (x) = 2x
(18.2b)
V 0 (x) = 1 , V 1 (x) = 2x − 1 ; W 0 (x) = 1 , W 1 (x) = 2x + 1. (18.2c)