Chebyshev Polynomials

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Pair of polynomial sequences
Not to be confused with discrete Chebyshev polynomials.

Plot of the first five Tn Chebyshev polynomials (first kind)
Plot of the first five Un Chebyshev polynomials (second kind)
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as

T

n

(
x
)

{\displaystyle T_{n}(x)}

and

U

n

(
x
)

{\displaystyle U_{n}(x)}

. They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind

T

n

{\displaystyle T_{n}}

are defined by

T

n

(
cos
⁡
θ
)
=
cos
⁡
(
n
θ
)
.

{\displaystyle T_{n}(\cos \theta )=\cos(n\theta ).}

Similarly, the Chebyshev polynomials of the second kind

U

n

{\displaystyle U_{n}}

are defined by

U

n

(
cos
⁡
θ
)
sin
⁡
θ
=

sin

(

(
n
+
1
)
θ

)

.

{\displaystyle U_{n}(\cos \theta )\sin \theta ={\sin }{\big (}(n+1)\theta {\big )}.}

That these expressions define polynomials in

cos
⁡
θ

{\displaystyle \cos \theta }

is not obvious at first sight but can be shown using de Moivre's formula (see below).
The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.[1]
In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;[2] the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature.
These polynomials were named after Pafnuty Chebyshev.[3] The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

Definitions[edit]
Recurrence definition[edit]
The Chebyshev polynomials of the first kind can be defined by the recurrence relation

T

0

(
x
)

=
1
,

T

1

(
x
)

=
x
,

T

n
+
1

(
x
)

=
2
x

T

n

(
x
)
−

T

n
−
1

(
x
)
.

{\displaystyle {\begin{aligned}T_{0}(x)&=1,\\T_{1}(x)&=x,\\T_{n+1}(x)&=2x\,T_{n}(x)-T_{n-1}(x).\end{aligned}}}

The Chebyshev polynomials of the second kind can be defined by the recurrence relation

U

0

(
x
)

=
1
,

U

1

(
x
)

=
2
x
,

U

n
+
1

(
x
)

=
2
x

U

n

(
x
)
−

U

n
−
1

(
x
)
,

{\displaystyle {\begin{aligned}U_{0}(x)&=1,\\U_{1}(x)&=2x,\\U_{n+1}(x)&=2x\,U_{n}(x)-U_{n-1}(x),\end{aligned}}}

which differs from the above only by the rule for n=1.

Trigonometric definition[edit]
The Chebyshev polynomials of the first and second kind can be defined as the unique polynomials satisfying

T

n

(
cos
⁡
θ
)
=
cos
⁡
(
n
θ
)

{\displaystyle T_{n}(\cos \theta )=\cos(n\theta )\quad }

and

U

n

(
cos
⁡
θ
)
=

sin

(

(
n
+
1
)
θ

)

sin
⁡
θ

,

{\displaystyle U_{n}(\cos \theta )={\frac {{\sin }{\big (}(n+1)\theta {\big )}}{\sin \theta }},}

for n = 0, 1, 2, 3, ….
An equivalent way to state this is via exponentiation of a complex number: given a complex number z = a + bi with absolute value of one,

z

n

=

T

n

(
a
)
+
i
b

U

n
−
1

(
a
)
.

{\displaystyle z^{n}=T_{n}(a)+ib\,U_{n-1}(a).}

Chebyshev polynomials can also be defined in this form when studying trigonometric polynomials.[4]
That

cos
⁡
(
n
x
)

{\displaystyle \cos(nx)}

is an

n

{\displaystyle n}

th-degree polynomial in

cos
⁡
(
x
)

{\displaystyle \cos(x)}

can be seen by observing that

cos
⁡
(
n
x
)

{\displaystyle \cos(nx)}

is the real part of one side of de Moivre's formula:

cos
⁡
n
θ
+
i
sin
⁡
n
θ
=
(
cos
⁡
θ
+
i
sin
⁡
θ

)

n

.

{\displaystyle \cos n\theta +i\sin n\theta =(\cos \theta +i\sin \theta )^{n}.}

The real part of the other side is a polynomial in

cos
⁡
x

{\displaystyle \cos x}

and

sin
⁡
x

{\displaystyle \sin x}

, in which all powers of

sin
⁡
x

{\displaystyle \sin x}

are even and thus replaceable through the identity

cos

2

⁡
x
+

sin

2

⁡
x
=
1

{\displaystyle \cos ^{2}x+\sin ^{2}x=1}

. By the same reasoning,

sin
⁡
n
x

{\displaystyle \sin nx}

is the imaginary part of the polynomial, in which all powers of

sin
⁡
x

{\displaystyle \sin x}

are odd and thus, if one factor of

sin
⁡
x

{\displaystyle \sin x}

is factored out, the remaining factors can be replaced to create a polynomial of degree

n
−
1

{\displaystyle n-1}

in

cos
⁡
x

{\displaystyle \cos x}

.
For

x

{\displaystyle x}

outside the interval [-1,1], the above definition implies

T

n

(
x
)
=

{

cos
⁡
(
n
arccos
⁡
x
)

 if 

|

x

|

≤
1
,

cosh
⁡
(
n
arccosh
⁡
x
)

 if 

x
≥
1
,

(
−
1

)

n

cosh

(
n
arccosh
⁡
(
−
x
)
)

 if 

x
≤
−
1.

{\displaystyle T_{n}(x)={\begin{cases}\cos(n\arccos x)&{\text{ if }}|x|\leq 1,\\\cosh(n\operatorname {arccosh} x)&{\text{ if }}x\geq 1,\\{(-1)^{n}\cosh }(n\operatorname {arccosh} (-x))&{\text{ if }}x\leq -1.\end{cases}}}

Commuting polynomials definition[edit]
Chebyshev polynomials can also be characterized by the following theorem:[5]
If

F

n

(
x
)

{\displaystyle F_{n}(x)}

is a family of monic polynomials with coefficients in a field of characteristic

0

{\displaystyle 0}

such that

deg
⁡

F

n

(
x
)
=
n

{\displaystyle \deg F_{n}(x)=n}

and

F

m

(

F

n

(
x
)

)

=

F

n

(

F

m

(
x
)

)

{\displaystyle F_{m}{\bigl (}F_{n}(x){\bigr )}=F_{n}{\bigl (}F_{m}(x){\bigr )}}

for all

m

{\displaystyle m}

and

n

{\displaystyle n}

, then, up to a simple change of variables, either

F

n

(
x
)
=

x

n

{\displaystyle F_{n}(x)=x^{n}}

for all

n

{\displaystyle n}

or

F

n

(
x
)
=
2
⋅

T

n

(

1
2

x

)

{\displaystyle F_{n}(x)=2\cdot T_{n}{\bigl (}{\tfrac {1}{2}}x{\bigr )}}

for all

n

{\displaystyle n}

.

Pell equation definition[edit]
The Chebyshev polynomials can also be defined as the solutions to the Pell equation:

T

n

(
x

)

2

−
(

x

2

−
1
)

U

n
−
1

(
x

)

2

=
1

{\displaystyle T_{n}(x)^{2}-(x^{2}-1)\,U_{n-1}(x)^{2}=1}

in a ring ⁠

R
[
x
]

{\displaystyle R[x]}

⁠.[6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

T

n

(
x
)
+

U

n
−
1

(
x
)

x

2

−
1

=

(

x
+

x

2

−
1

 

)

n

.

{\displaystyle T_{n}(x)+U_{n-1}(x){\textstyle {\sqrt {x^{2}-1}}}={\bigl (}{\textstyle x+{\sqrt {x^{2}-1}}~\!}{\bigr )}^{n}.}

Generating functions[edit]
The ordinary generating function for

T

n

{\displaystyle T_{n}}

is

∑

n
=
0

∞

T

n

(
x
)

t

n

=

1
−
t
x

1
−
2
t
x
+

t

2

.

{\displaystyle \sum _{n=0}^{\infty }T_{n}(x)\,t^{n}={\frac {1-tx}{1-2tx+t^{2}}}.}

There are several other generating functions for the Chebyshev polynomials; the exponential generating function is

∑

n
=
0

∞

T

n

(
x
)

t

n

n
!

=

1
2

(

exp

(

t

(

x
−

x

2

−
1

 

)

)

+

exp

(

t

(

x
+

x

2

−
1

 

)

)

)

=

e

t
x

cosh

(

t

x

2

−
1

 

)

.

{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }T_{n}(x){\frac {t^{n}}{n!}}&={\tfrac {1}{2}}{\Bigl (}{\exp }{\Bigl (}t{\bigl (}{\textstyle x-{\sqrt {x^{2}-1}}~\!}{\bigr )}{\Bigr )}+{\exp }{\Bigl (}t{\bigl (}{\textstyle x+{\sqrt {x^{2}-1}}~\!}{\bigr )}{\Bigr )}{\Bigr )}\\&={e^{tx}\cosh }{\bigl (}{\textstyle t{\sqrt {x^{2}-1}}}~\!{\bigr )}.\end{aligned}}}

The generating function relevant for 2-dimensional potential theory and multipole expansion is

∑

n
=
1

∞

T

n

(
x
)

t

n

n

=
ln
⁡

(

1

1
−
2
t
x
+

t

2

)

.

{\displaystyle \sum \limits _{n=1}^{\infty }T_{n}(x){\frac {t^{n}}{n}}=\ln \left({\frac {1}{\sqrt {1-2tx+t^{2}}}}\right).}

The ordinary generating function for Un is

∑

n
=
0

∞

U

n

(
x
)

t

n

=

1

1
−
2
t
x
+

t

2

,

{\displaystyle \sum _{n=0}^{\infty }U_{n}(x)\,t^{n}={\frac {1}{1-2tx+t^{2}}},}

and the exponential generating function is

∑

n
=
0

∞

U

n

(
x
)

t

n

n
!

=

e

t
x

(

cosh

(

t

x

2

−
1

 

)

+

x

x

2

−
1

sinh

(

t

x

2

−
1

 

)

)

.

{\displaystyle \sum _{n=0}^{\infty }U_{n}(x){\frac {t^{n}}{n!}}=e^{tx}{\biggl (}{\cosh }{\bigl (}{\textstyle t{\sqrt {x^{2}-1}}~\!}{\bigr )}+{{\frac {x}{\sqrt {x^{2}-1}}}\sinh }{\bigl (}{\textstyle t{\sqrt {x^{2}-1}}~\!}{\bigr )}{\biggr )}.}

Relations between the two kinds of Chebyshev polynomials[edit]
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences

V
~

n

(
P
,
Q
)

{\displaystyle {\tilde {V}}_{n}(P,Q)}

and

U
~

n

(
P
,
Q
)

{\displaystyle {\tilde {U}}_{n}(P,Q)}

with parameters

P
=
2
x

{\displaystyle P=2x}

and

Q
=
1

{\displaystyle Q=1}

:

U
~

n

(
2
x
,
1
)

=

U

n
−
1

(
x
)
,

V
~

n

(
2
x
,
1
)

=
2

T

n

(
x
)
.

{\displaystyle {\begin{aligned}{\tilde {U}}_{n}(2x,1)&=U_{n-1}(x),\\{\tilde {V}}_{n}(2x,1)&=2\,T_{n}(x).\end{aligned}}}

It follows that they also satisfy a pair of mutual recurrence equations:[7]

T

n
+
1

(
x
)

=
x

T

n

(
x
)
−
(
1
−

x

2

)

U

n
−
1

(
x
)
,

U

n
+
1

(
x
)

=
x

U

n

(
x
)
+

T

n
+
1

(
x
)
.

{\displaystyle {\begin{aligned}T_{n+1}(x)&=x\,T_{n}(x)-(1-x^{2})\,U_{n-1}(x),\\U_{n+1}(x)&=x\,U_{n}(x)+T_{n+1}(x).\end{aligned}}}

The second of these may be rearranged using the recurrence definition for the Chebyshev polynomials of the second kind to give:

T

n

(
x
)
=

1
2

(

U

n

(
x
)
−

U

n
−
2

(
x
)

)

.

{\displaystyle T_{n}(x)={\tfrac {1}{2}}{\big (}U_{n}(x)-U_{n-2}(x){\big )}.}

Using this formula iteratively gives the sum formula:

U

n

(
x
)
=

{

2

∑

 odd 

j
>
0

n

T

j

(
x
)

 for odd 

n
.

2

∑

 even 

j
≥
0

n

T

j

(
x
)
−
1

 for even 

n
,

{\displaystyle U_{n}(x)={\begin{cases}2\sum _{{\text{ odd }}j>0}^{n}T_{j}(x)&{\text{ for odd }}n.\\2\sum _{{\text{ even }}j\geq 0}^{n}T_{j}(x)-1&{\text{ for even }}n,\