# Chebyshev polynomials

Template:Distinguish In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

The Chebyshev polynomials Tn or Un are polynomials of degree Template:Mvar and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.

Chebyshev polynomials are polynomials with the largest possible leading coefficient, but subject to the condition that their absolute value on the interval Template:Closed-closed is bounded by 1. They are also the extremal polynomials for many other properties.[2]

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw–Curtis quadrature.

In the study of differential equations they arise as the solution to the Chebyshev differential equations

${\displaystyle (1-x^{2})\,y''-x\,y'+n^{2}\,y=0\,\!}$

and

${\displaystyle (1-x^{2})\,y''-3x\,y'+n(n+2)\,y=0\,\!}$

for the polynomials of the first and second kind, respectively. These equations are special cases of the Sturm–Liouville differential equation.

## Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation

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

The ordinary generating function for Tn is

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

the exponential generating function is

${\displaystyle \sum _{n=0}^{\infty }T_{n}(x){\frac {t^{n}}{n!}}={\tfrac {1}{2}}\left(e^{(x-{\sqrt {x^{2}-1}})t}+e^{(x+{\sqrt {x^{2}-1}})t}\right)=e^{tx}\cosh(t{\sqrt {x^{2}-1}}).\,\!}$

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

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

The Chebyshev polynomials of the second kind are defined by the recurrence relation

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

The ordinary generating function for Un is

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

the exponential generating function is

${\displaystyle \sum _{n=0}^{\infty }U_{n}(x){\frac {t^{n}}{n!}}=e^{tx}\left(\cosh(t{\sqrt {x^{2}-1}})+{\frac {x}{\sqrt {x^{2}-1}}}\sinh(t{\sqrt {x^{2}-1}})\right).\,\!}$

### Trigonometric definition

The Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying

${\displaystyle T_{n}(x)=\cos(n\arccos x)=\cosh(n\,\operatorname {arcosh} \,x)\,\!}$

or, in other words, as the unique polynomials satisfying

${\displaystyle T_{n}(\cos(\vartheta ))=\cos(n\vartheta )\,\!}$

for n = 0, 1, 2, 3, ... which is a variant (equivalent transpose) of Schröder's equation, viz. Tn(x) is functionally conjugate to nx, codified in the nesting property below. Further compare to the spread polynomials, in the section below.

The polynomials of the second kind satisfy:

${\displaystyle U_{n}(\cos(\vartheta ))={\frac {\sin((n+1)\vartheta )}{\sin \vartheta }}\,,}$

which is structurally quite similar to the Dirichlet kernel ${\displaystyle D_{n}(x)\,\!}$:

${\displaystyle D_{n}(x)={\frac {\sin((2n+1)(x/2))}{\sin(x/2)}}=U_{2n}(\cos(x/2))\,.}$

That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of de Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable through the identity cos2(x) + sin2(x) = 1.

This identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle.

Evaluating the first two Chebyshev polynomials,

${\displaystyle T_{0}(x)=\cos(0x)=1}$

and

${\displaystyle T_{1}(\cos(x))=\cos(x)\,,}$

one can straightforwardly determine that

${\displaystyle \cos(2\vartheta )=2\cos \vartheta \cos \vartheta -\cos(0\vartheta )=2\cos ^{2}\,\vartheta -1\,\!}$
${\displaystyle \cos(3\vartheta )=2\cos \vartheta \cos(2\vartheta )-\cos \vartheta =4\cos ^{3}\,\vartheta -3\cos \vartheta \,,}$

and so forth.

Two immediate corollaries are the composition identity (or nesting property specifying a semigroup)

${\displaystyle T_{n}(T_{m}(x))=T_{nm}(x)\,;}$

and the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,

{\displaystyle {\begin{aligned}z^{n}&=|z|^{n}\left(\cos \left(n\arccos {\frac {a}{|z|}}\right)+i\sin \left(n\arccos {\frac {a}{|z|}}\right)\right)\\&=|z|^{n}T_{n}\left({\frac {a}{|z|}}\right)+ib\ |z|^{n-1}\ U_{n-1}\left({\frac {a}{|z|}}\right).\end{aligned}}}

### Pell equation definition

The Chebyshev polynomials can also be defined as the solutions to the Pell equation

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

in a ring R[x].[3] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

${\displaystyle T_{n}(x)+U_{n-1}(x){\sqrt {x^{2}-1}}=(x+{\sqrt {x^{2}-1}})^{n}.\,\!}$

### Products of Chebyshev polynomials

When working with Chebyshev polynomials quite often products of two of them occur. These products can be reduced to combinations of Chebyshev polynomials with lower or higher degree and concluding statements about the product are easier to make. It shall be assumed that in the following the index m is greater than or equal to the index n and n is not negative. For Chebyshev polynomials of the first kind the product expands to

${\displaystyle 2T_{m}(x)T_{n}(x)=T_{m+n}(x)+T_{m-n}(x)}$

which is an analogy to the addition theorem ${\displaystyle 2\cos \alpha \cos \beta =\cos(\alpha +\beta )+\cos(\alpha -\beta )}$ with the identities ${\displaystyle \alpha =m\arccos x,\beta =n\arccos x.}$ For n=1 this results in the already known recurrence formula, just arranged differently, and with n=2 it forms the recurrence relation for all even or all odd Chebyshev polynomials (depending on the parity of the lowest m) which allows to design functions with prescribed symmetry properties. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:

${\displaystyle T_{2n}(x)=2T_{n}^{2}(x)-T_{0}(x)=2T_{n}^{2}(x)-1}$
${\displaystyle T_{2n+1}(x)=2T_{n+1}(x)T_{n}(x)-T_{1}(x)=2T_{n+1}(x)T_{n}(x)-x}$
${\displaystyle T_{2n-1}(x)=2T_{n-1}(x)T_{n}(x)-T_{1}(x)=2T_{n-1}(x)T_{n}(x)-x}$

For Chebyshev polynomials of the second kind products may be written as:

${\displaystyle U_{m}(x)U_{n}(x)=\sum _{k=0}^{n}U_{m-n+2k}(x)=\sum _{p=m-n,step2}^{m+n}U_{p}(x).}$

By this, like above, with n=2 the recurrence formula of Chebyshev polynomials of the second kind forms for both types of symmetry to

${\displaystyle U_{m+2}(x)=U_{2}(x)U_{m}(x)-U_{m}(x)-U_{m-2}(x)=U_{m}(x)(U_{2}(x)-1)-U_{m-2}(x),}$ depending on whether m starts with 2 or 3.

## Relation between Chebyshev polynomials of the first and second kinds

The Chebyshev polynomials of the first and second kind are closely related by the following equations

${\displaystyle {\tfrac {d}{dx}}\,T_{n}(x)=nU_{n-1}(x){\mbox{ , }}n=1,\ldots }$
${\displaystyle T_{n}(x)={\tfrac {1}{2}}(U_{n}(x)-\,U_{n-2}(x)).}$
${\displaystyle T_{n+1}(x)=xT_{n}(x)-(1-x^{2})U_{n-1}(x)\,}$
${\displaystyle T_{n}(x)=U_{n}(x)-x\,U_{n-1}(x),}$
${\displaystyle U_{n}(x)=2\sum _{j\,\,{\text{odd}}}^{n}T_{j}(x)}$, where n is odd.
${\displaystyle U_{n}(x)=2\sum _{j\,{\text{even}}}^{n}T_{j}(x)-1}$, where n is even.

The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations

${\displaystyle 2T_{n}(x)={\frac {1}{n+1}}\;{\frac {d}{dx}}T_{n+1}(x)-{\frac {1}{n-1}}\;{\frac {d}{dx}}T_{n-1}(x){\mbox{ , }}\quad n=1,\ldots }$

This relationship is used in the Chebyshev spectral method of solving differential equations.

Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:

${\displaystyle T_{0}(x)=1\,\!}$
${\displaystyle U_{-1}(x)=0\,\!}$
${\displaystyle T_{n+1}(x)=xT_{n}(x)-(1-x^{2})U_{n-1}(x)\,}$
${\displaystyle U_{n}(x)=xU_{n-1}(x)+T_{n}(x)\,}$

These can be derived from the trigonometric formulae; for example, if ${\displaystyle x=\cos \vartheta }$, then

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

Note that both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our Un (the polynomial of degree n) with Un+1 instead.

Turán's inequalities for the Chebyshev polynomials are

${\displaystyle T_{n}(x)^{2}-T_{n-1}(x)T_{n+1}(x)=1-x^{2}>0{\text{ for }}-1 and
${\displaystyle U_{n}(x)^{2}-U_{n-1}(x)U_{n+1}(x)=1>0.\!}$

The integral relations are

${\displaystyle \int _{-1}^{1}{\frac {T_{n}(y)dy}{(y-x){\sqrt {1-y^{2}}}}}=\pi U_{n-1}(x),}$
${\displaystyle \int _{-1}^{1}{\frac {{\sqrt {1-y^{2}}}U_{n-1}(y)dy}{y-x}}=-\pi T_{n}(x)}$

where integrals are considered as principal value.

## Explicit expressions

Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:

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

${\displaystyle \cos \left((2k+1)\,{\tfrac {\pi }{2}}\right)=0}$