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# Chebyshev polynomial

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### Chebyshev polynomial

Not to be confused with discrete Chebyshev polynomials.

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 is bounded on the interval 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

$\left(1-x^2\right)\,y$ - x\,y' + n^2\,y = 0 \,\!

and

$\left(1-x^2\right)\,y$ - 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



\begin{align} T_0(x) & = 1 \\ T_1(x) & = x \\ T_{n+1}(x) & = 2xT_n(x) - T_{n-1}(x). \end{align}

The conventional generating function for Tn is

$\sum_\left\{n=0\right\}^\left\{\infty\right\}T_n\left(x\right) t^n = \frac\left\{1-tx\right\}\left\{1-2tx+t^2\right\}. \,\!$

The exponential generating function is

$\sum_\left\{n=0\right\}^\left\{\infty\right\}T_n\left(x\right) \frac\left\{t^n\right\}\left\{n!\right\} = \tfrac\left\{1\right\}\left\{2\right\}\left\left( e^\left\{\left(x-\sqrt\left\{x^2 -1\right\}\right)t\right\}+e^\left\{\left(x+\sqrt\left\{x^2 -1\right\}\right)t\right\}\right\right). \,\!$

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

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

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



\begin{align} U_0(x) & = 1 \\ U_1(x) & = 2x \\ U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x). \end{align}

One example of a generating function for Un is

$\sum_\left\{n=0\right\}^\left\{\infty\right\}U_n\left(x\right) t^n = \frac\left\{1\right\}\left\{1-2 t x+t^2\right\}. \,\!$

### Trigonometric definition

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

$T_n\left(x\right)=\cos\left(n \arccos x\right)=\cosh\left(n\,\mathrm\left\{arccosh\right\}\,x\right) \,\!$

or, in other words, as the unique polynomials satisfying

$T_n\left(\cos\left(\vartheta\right)\right)=\cos\left(n\vartheta\right) \,\!$

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:

$U_n\left(\cos\left(\vartheta\right)\right) = \frac\left\{\sin\left(\left(n+1\right)\vartheta\right)\right\}\left\{\sin\vartheta\right\} \, ,$

which is structurally quite similar to the Dirichlet kernel $D_n\left(x\right) \,\!$:

$D_n\left(x\right) = \frac\left\{\sin\left(\left(2n+1\right)\left(x/2\right)\right)\right\}\left\{\sin \left(x/2\right)\right\} = U_\left\{2n\right\}\left(\cos \left(x/2\right)\right)\, .$

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,

$T_0\left(x\right)=\cos\left(0x\right) =1$

and

$T_1\left(\cos\left(x\right)\right)=\cos\left(x\right) \, ,$

one can straightforwardly determine that



\cos(2 \vartheta)=2\cos\vartheta \cos\vartheta - \cos(0 \vartheta) = 2\cos^{2}\,\vartheta - 1 \,\!



\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)

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



\begin{align} 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{align}

### Pell equation definition

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

$T_n\left(x\right)^2 - \left(x^2-1\right) U_\left\{n-1\right\}\left(x\right)^2 = 1 \,\!$

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:

$T_n\left(x\right) + U_\left\{n-1\right\}\left(x\right) \sqrt\left\{x^2-1\right\} = \left(x + \sqrt\left\{x^2-1\right\}\right)^n. \,\!$

## 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

$\tfrac\left\{d\right\}\left\{dx\right\} \, T_n\left(x\right) = n U_\left\{n-1\right\}\left(x\right) \mbox\left\{ , \right\} n=1,\ldots$
$T_n\left(x\right) = \tfrac\left\{1\right\}\left\{2\right\} \left(U_n\left(x\right) - \, U_\left\{n-2\right\}\left(x\right)\right).$
$T_\left\{n+1\right\}\left(x\right) = xT_n\left(x\right) - \left(1 - x^2\right)U_\left\{n-1\right\}\left(x\right)\,$
$T_n\left(x\right) = U_n\left(x\right) - x \, U_\left\{n-1\right\}\left(x\right),$
$U_n\left(x\right) =2\sum_\left\{j\,\, \text\left\{odd\right\}\right\}^n T_j\left(x\right)$, where n is odd.
$U_n\left(x\right) =2\sum_\left\{j\, \text\left\{even\right\}\right\}^n T_j\left(x\right)-1$, where n is even.

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

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

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:

$T_0\left(x\right) = 1\,\!$
$U_\left\{-1\right\}\left(x\right) = 0\,\!$
$T_\left\{n+1\right\}\left(x\right) = xT_n\left(x\right) - \left(1 - x^2\right)U_\left\{n-1\right\}\left(x\right)\,$
$U_n\left(x\right) = xU_\left\{n-1\right\}\left(x\right) + T_n\left(x\right)\,$

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

\begin\left\{align\right\}
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{align}

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

$U_n\left(x\right)^2-U_\left\{n-1\right\}\left(x\right) U_\left\{n+1\right\}\left(x\right)= 1>0.\!$

## Explicit expressions

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

$T_n\left(x\right) =$

\begin{cases} \cos(n\arccos(x)), & \ |x| \le 1 \\ \cosh(n \, \mathrm{arccosh}(x)), & \ x \ge 1 \\ (-1)^n \cosh(n \, \mathrm{arccosh}(-x)), & \ x \le -1 \\ \end{cases} \,\!



\begin{align} T_n(x) & = \frac{(x-\sqrt{x^2-1})^n+(x+\sqrt{x^2-1})^n}{2} \\ & = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (x^2-1)^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (1 - x^{-2})^k \\ & = \tfrac{n}{2}\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} \quad (n>0) \\ & = n \sum_{k=0}^{n}(-2)^{k} \frac{(n+k-1)!} {(n-k)!(2k)!}(1 - x)^k \quad (n>0)\\ & = \, _2F_1\left(-n,n;\frac 1 2; \frac{1-x} 2 \right) \\ \end{align}



\begin{align} U_n(x) & = \frac{(x+\sqrt{x^2-1})^{n+1} - (x-\sqrt{x^2-1})^{n+1}}{2\sqrt{x^2-1}} \\ & = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (x^2-1)^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (1 - x^{-2})^k \\ & =\sum_{k=0}^{\lfloor n/2\rfloor} \binom{2k-(n+1)}{k}~(2x)^{n-2k} \quad (n>0)\\ & =\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k \binom{n-k}{k}~(2x)^{n-2k} \quad (n>0)\\ & = \sum_{k=0}^{n}(-2)^{k} \frac{(n+k+1)!} {(n-k)!(2k+1)!}(1 - x)^k \quad (n>0)\\ & = (n+1) \, _2F_1\left(-n,n+2; \tfrac{3}{2}; \tfrac{1}{2}\left[1-x\right] \right) \end{align}

where $_2F_1$ is a hypergeometric function.

## Properties

### Roots and extrema

A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1,1]. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that

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

one can easily prove that the roots of Tn are

$x_k = \cos\left\left(\tfrac\left\{\pi\right\}\left\{2\right\}\,\frac\left\{2k-1\right\}\left\{n\right\}\right\right),\quad k=1,\ldots,n.$

Similarly, the roots of Un are

$x_k = \cos\left\left(\frac\left\{k\right\}\left\{n+1\right\}\pi\right\right),\quad k=1,\ldots,n.$

The extrema of Tn on the interval −1 ≤ x ≤ 1 are located at

$x_k = \cos\left\left(\tfrac\left\{k\right\}\left\{n\right\}\pi\right\right),\quad k=0,\ldots,n.$

One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:

$T_n\left(1\right) = 1\,$
$T_n\left(-1\right) = \left(-1\right)^n\,$
$U_n\left(1\right) = n + 1\,$
$U_n\left(-1\right) = \left(n + 1\right)\left(-1\right)^n.\,$

### Differentiation and integration

The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it's easy to show that:

$\frac\left\{d T_n\right\}\left\{d x\right\} = n U_\left\{n - 1\right\}\,$
$\frac\left\{d U_n\right\}\left\{d x\right\} = \frac\left\{\left(n + 1\right)T_\left\{n + 1\right\} - x U_n\right\}\left\{x^2 - 1\right\}\,$
$\frac\left\{d^2 T_n\right\}\left\{d x^2\right\} = n \frac\left\{n T_n - x U_\left\{n - 1\right\}\right\}\left\{x^2 - 1\right\} = n \frac\left\{\left(n + 1\right)T_n - U_n\right\}\left\{x^2 - 1\right\}.\,$

The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:

$\frac\left\{d^2 T_n\right\}\left\{d x^2\right\} \Bigg|_\left\{x = 1\right\} \!\! = \frac\left\{n^4 - n^2\right\}\left\{3\right\},$
$\frac\left\{d^2 T_n\right\}\left\{d x^2\right\} \Bigg|_\left\{x = -1\right\} \!\! = \left(-1\right)^n \frac\left\{n^4 - n^2\right\}\left\{3\right\}.$
Proof

The second derivative of the Chebyshev polynomial of the first kind is

$T$_n = n \frac{n T_n - x U_{n - 1}}{x^2 - 1}

which, if evaluated as shown above, poses a problem because it is indeterminate at x = ±1. Since the function is a polynomial, (all of) the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:

$T$_n(1) = \lim_{x \to 1} n \frac{n T_n - x U_{n - 1}}{x^2 - 1}

where only $x = 1$ is considered for now. Factoring the denominator:

$T$_n(1) = \lim_{x \to 1} n \frac{n T_n - x U_{n - 1}}{(x + 1)(x - 1)} =

\lim_{x \to 1} n \frac{\frac{n T_n - x U_{n - 1}}{x - 1}}{x + 1}.

Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and

$T$_n(1) = n \frac{\lim_{x \to 1} \frac{n T_n - x U_{n - 1}}{x - 1}}{\lim_{x \to 1} (x + 1)} =

\frac{n}{2} \lim_{x \to 1} \frac{n T_n - x U_{n - 1}}{x - 1} .

The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e. $U_\left\{n - 1\right\}\left(1\right) = n T_n\left(1\right) = n$ which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:

\begin\left\{align\right\}
T_n(1) & = \frac{n}{2} \lim_{x \to 1} \frac{\frac{d}{dx}(n T_n - x U_{n - 1})}{\frac{d}{dx}(x - 1)} \\
& = \frac{n}{2} \lim_{x \to 1} \frac{d}{dx}(n T_n - x U_{n - 1}) \\
& = \frac{n}{2} \lim_{x \to 1} \left(n^2 U_{n - 1} - U_{n - 1} - x \frac{d}{dx}(U_{n - 1})\right) \\
& = \frac{n}{2} \left(n^2 U_{n - 1}(1) - U_{n - 1}(1) - \lim_{x \to 1} x \frac{d}{dx}(U_{n - 1})\right) \\
& = \frac{n^4}{2} - \frac{n^2}{2} - \frac{1}{2} \lim_{x \to 1} \frac{d}{dx}(n U_{n - 1}) \\
& = \frac{n^4}{2} - \frac{n^2}{2} - \frac{T_n(1)}{2} \\
T_n(1) & = \frac{n^4 - n^2}{3}. \\


\end{align}

The proof for $x = -1$ is similar, with the fact that $T_n\left(-1\right) = \left(-1\right)^n$ being important.

Indeed, the following, more general formula holds:

$\frac\left\{d^p T_n\right\}\left\{d x^p\right\} \Bigg|_\left\{x = \pm 1\right\} \!\! = \left(\pm 1\right)^\left\{n+p\right\}\prod_\left\{k=0\right\}^\left\{p-1\right\}\frac\left\{n^2-k^2\right\}\left\{2k+1\right\}.$

This latter result is of great use in the numerical solution of eigenvalue problems.

Concerning integration, the first derivative of the Tn implies that

$\int U_n\, dx = \frac\left\{T_\left\{n + 1\right\}\right\}\left\{n + 1\right\}\,$

and the recurrence relation for the first kind polynomials involving derivatives establishes that

$\int T_n\, dx = \frac\left\{1\right\}\left\{2\right\} \left\left(\frac\left\{T_\left\{n + 1\right\}\right\}\left\{n + 1\right\} - \frac\left\{T_\left\{n - 1\right\}\right\}\left\{n - 1\right\}\right\right) = \frac\left\{n T_\left\{n + 1\right\}\right\}\left\{n^2 - 1\right\} - \frac\left\{x T_n\right\}\left\{n - 1\right\}.\,$

### Orthogonality

Both the Tn and the Un form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight

$\frac\left\{1\right\}\left\{\sqrt\left\{1-x^2\right\}\right\}, \,\!$

on the interval [−1,1], i.e. we have:

$\int_\left\{-1\right\}^1 T_n\left(x\right)T_m\left(x\right)\,\frac\left\{dx\right\}\left\{\sqrt\left\{1-x^2\right\}\right\}=$

\begin{cases} 0 &: n\ne m \\ \pi &: n=m=0\\ \pi/2 &: n=m\ne 0 \end{cases}

This can be proven by letting x = cos (θ) and using the defining identity Tn(cos (θ)) = cos ().

Similarly, the polynomials of the second kind are orthogonal with respect to the weight

$\sqrt\left\{1-x^2\right\} \,\!$

on the interval [−1,1], i.e. we have:

$\int_\left\{-1\right\}^1 U_n\left(x\right)U_m\left(x\right)\sqrt\left\{1-x^2\right\}\,dx =$

\begin{cases} 0 &: n\ne m, \\ \pi/2 &: n=m. \end{cases}

(Note that the measure $\sqrt\left\{1-x^2\right\} \,dx$ is, to within a normalizing constant, the Wigner semicircle distribution).

The Tn also satisfy a discrete orthogonality condition:

$\sum_\left\{k=0\right\}^\left\{N-1\right\}\left\{T_i\left(x_k\right)T_j\left(x_k\right)\right\} =$

\begin{cases} 0 &: i\ne j \\ N &: i=j=0 \\ N/2 &: i=j\ne 0 \end{cases} \,\!

where the xk are the N GaussLobatto zeros of TN(x)

$x_k=\cos\left\left(\frac\left\{\pi\left\left(k+\frac\left\{1\right\}\left\{2\right\}\right\right)\right\}\left\{N\right\}\right\right) .$

### Minimal ∞-norm

For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1,

$f\left(x\right) = \frac1\left\{2^\left\{n-1\right\}\right\}T_n\left(x\right)$

is the one of which the maximal absolute value on the interval [−1, 1] is minimal.

This maximal absolute value is

$\frac1\left\{2^\left\{n-1\right\}\right\}$

and |ƒ(x)| reaches this maximum exactly n + 1 times at

$x = \cos \frac\left\{k\pi\right\}\left\{n\right\}\text\left\{ for \right\}0 \le k \le n.$
Proof

Let's assume that $w_n\left(x\right)$ is a polynomial of degree n with leading coefficient 1 with maximal absolute value on the interval [−1, 1] less than $\frac1\left\{2^\left\{n-1\right\}\right\}$.

Define

$f_n\left(x\right) = \frac1\left\{2^\left\{n-1\right\}\right\}T_n\left(x\right) - w_n\left(x\right)$

Because at extreme points of $T_n$ we have $|w_n\left(x\right)| < \left|\frac1\left\{2^\left\{n-1\right\}\right\}T_n\left(x\right)\right|$

$f_n\left(x\right) > 0 \text\left\{ for \right\} x = \cos \frac\left\{2k\pi\right\}\left\{n\right\} \text\left\{ where \right\} 0 \le 2k \le n$
$f_n\left(x\right) < 0 \text\left\{ for \right\} x = \cos \frac\left\{\left(2k + 1\right)\pi\right\}\left\{n\right\} \text\left\{ where \right\} 0 \le 2k + 1 \le n$

From the intermediate value theorem, $f_n\left(x\right)$ has at least n roots. However, this is impossible, as $f_n\left(x\right)$ is a polynomial of degree n − 1, so the fundamental theorem of algebra implies it has at most n − 1 roots.

### Other properties

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials:

• $T_n\left(x\right)= \frac 1$Template:N-\frac 1 2 \choose n P_n^{-\frac 1 2, -\frac 1 2}(x)= \frac n 2 C_n^0(x),
• $U_n\left(x\right)= \frac 1\left\{2\left\{n+\frac 1 2 \choose n\right\}\right\} P_n^\left\{\frac 1 2, \frac 1 2\right\}\left(x\right)= C_n^1\left(x\right).$

For every nonnegative integer n, Tn(x) and Un(x) are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of x, it only has even or odd degree terms respectively. In fact,

$T_\left\{2n\right\}\left(x\right)=T_n\left\left(2x^2-1\right\right)=2 T_n\left(x\right)^2-1$

and

$2 x U_n\left\left(1-2x^2\right\right)= \left(-1\right)^n U_\left\{2n+1\right\}\left(x\right).$

The leading coefficient of Tn is 2n − 1 if 1 ≤ n, but 1 if 0 = n.

Tn are a special case of Lissajous curves with frequency ratio equal to n.

Several polynomial sequences like Lucas polynomials (Ln), Dickson polynomials(Dn), Fibonacci polynomials(Fn) are related to Chebyshev polynomials Tn and Un.

The Chebyshev polynomials of the first kind satisfy the relation

$T_j\left(x\right) T_k\left(x\right) = \tfrac\left\{1\right\}\left\{2\right\}\left\left( T_\left\{j+k\right\}\left(x\right) + T_\left\{|k-j|\right\}\left(x\right)\right\right),\quad\forall j,k\ge 0,\,$

which is easily proved from the product-to-sum formula for the cosine. The polynomials of the second kind satisfy the similar relation

$T_j\left(x\right) U_k\left(x\right) = \tfrac\left\{1\right\}\left\{2\right\}\left\left( U_\left\{j+k\right\}\left(x\right) + U_\left\{k-j\right\}\left(x\right)\right\right),\quad\forall j,k.$

Similar to the formula

$T_n\left\left(\cos\theta\right\right) = \cos\left(n \theta\right),$

we have the analogous formula

$T_\left\{2n+1\right\}\left\left(\sin\theta\right\right) = \left(-1\right)^n \sin\left(\left(2n+1\right)\theta\right)$.

For $x\ne 0$,

$T_n\left\left(\tfrac\left\{1\right\}\left\{2\right\}\left\left[x+x^\left\{-1\right\}\right\right]\right\right)=\tfrac\left\{1\right\}\left\{2\right\}\left\left(x^n+x^\left\{-n\right\}\right\right)$ and
$x^n= T_n\left\left(\tfrac\left\{1\right\}\left\{2\right\}\left\left[x+x^\left\{-1\right\}\right\right]\right\right)+ \tfrac\left\{1\right\}\left\{2\right\}\left\left(x-x^\left\{-1\right\}\right\right) U_\left\{n-1\right\}\left\left(\tfrac\left\{1\right\}\left\{2\right\}\left\left[x+x^\left\{-1\right\}\right\right]\right\right)$,

which follows from the fact that this holds by definition for $x = e^\left\{i\theta\right\}$.

Let

$C_n\left(x\right)=2T_n \left\left(\frac\left\{x\right\}\left\{2\right\}\right\right)$.

Then $C_n\left(x\right)$ and $C_m\left(x\right)$ are commuting polynomials:

$C_n\left\left(C_m\left(x\right)\right\right)=C_m\left(C_n\left(x\right)\right)$,

as is evident in the Abelian nesting property specified above.

## Examples

The first few Chebyshev polynomials of the first kind are A028297

$T_0\left(x\right) = 1 \,$
$T_1\left(x\right) = x \,$
$T_2\left(x\right) = 2x^2 - 1 \,$
$T_3\left(x\right) = 4x^3 - 3x \,$
$T_4\left(x\right) = 8x^4 - 8x^2 + 1 \,$
$T_5\left(x\right) = 16x^5 - 20x^3 + 5x \,$
$T_6\left(x\right) = 32x^6 - 48x^4 + 18x^2 - 1 \,$
$T_7\left(x\right) = 64x^7 - 112x^5 + 56x^3 - 7x \,$
$T_8\left(x\right) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \,$
$T_9\left(x\right) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \,$
$T_\left\{10\right\}\left(x\right) = 512x^\left\{10\right\} - 1280x^8 + 1120x^6 - 400x^4 + 50x^2-1. \,$
$T_\left\{11\right\}\left(x\right) = 1024x^\left\{11\right\} - 2816x^9 + 2816x^7 - 1232x^5 +220x^3 - 11x. \,$

The first few Chebyshev polynomials of the second kind are

$U_0\left(x\right) = 1 \,$
$U_1\left(x\right) = 2x \,$
$U_2\left(x\right) = 4x^2 - 1 \,$
$U_3\left(x\right) = 8x^3 - 4x \,$
$U_4\left(x\right) = 16x^4 - 12x^2 + 1 \,$
$U_5\left(x\right) = 32x^5 - 32x^3 + 6x \,$
$U_6\left(x\right) = 64x^6 - 80x^4 + 24x^2 - 1 \,$
$U_7\left(x\right) = 128x^7 - 192x^5 + 80x^3 - 8x \,$
$U_8\left(x\right) = 256x^8 - 448 x^6 + 240 x^4 - 40 x^2 + 1 \,$
$U_9\left(x\right) = 512x^9 - 1024 x^7 + 672 x^5 - 160 x^3 + 10 x. \,$

## As a basis set

In the appropriate Sobolev space, the set of Chebyshev polynomials form a orthonormal basis, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:[4]

$f\left(x\right) = \sum_\left\{n = 0\right\}^\infty a_n T_n\left(x\right).$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients an can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.

Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart.[4] These attributes include:

• The Chebyshev polynomials form a complete orthogonal system.
• The Chebyshev series converges to ƒ(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases — as long as there are a finite number of discontinuities in ƒ(x) and its derivatives.
• At a discontinuity, the series will converge to the average of the right and left limits.

The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method,[4] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

### Example 1

Consider the Chebyshev expansion of $\log\left(1+x\right)$. One can express

$\log\left(1+x\right) = \sum_\left\{n = 0\right\}^\infty a_n T_n\left(x\right).$

One can find the coefficients $a_n$ either through the application of an inner product or by the discrete orthogonality condition. For the inner product,



\int_{-1}^{+1}\frac{T_m(x)\log(1+x)}{\sqrt{1-x^2}}dx = \sum_{n=0}^{\infty}a_n\int_{-1}^{+1}\frac{T_m(x)T_n(x)}{\sqrt{1-x^2}}dx,

which gives



a_n= \begin{cases} -\log(2) &:n = 0 \\ \frac{-2(-1)^n}{n} &: n > 0. \end{cases}

Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discrete orthogonality condition gives



a_n=\frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}T_n(x_k)\log(1+x_k),

where $\delta_\left\{ij\right\}$ is the Kronecker delta function and the $x_k$ are the N Gauss–Lobatto zeros of $T_N\left(x\right)$

$x_k=\cos\left\left(\frac\left\{\pi\left\left(k+\frac\left\{1\right\}\left\{2\right\}\right\right)\right\}\left\{N\right\}\right\right) .$

This allows us to compute the coefficients $a_n$ very efficiently through the discrete cosine transform



a_n=\frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}\cos\left(\frac{n\pi\left(k+\frac{1}{2}\right)}{N}\right)\log(1+x_k).

### Example 2

To provide another example:

\begin\left\{align\right\}\left(1-x^2\right)^\alpha=& -\frac 1 \left\{\sqrt \pi\right\}\frac\left\{\Gamma\left(\frac 1 2+\alpha\right)\right\}\left\{\Gamma\left(\alpha+1\right)\right\}+ 2^\left\{1-2\alpha\right\} \sum_\left\{n=0\right\} \left(-1\right)^n \left\{2\alpha \choose \alpha-n\right\} T_\left\{2n\right\}\left(x\right)\\=& 2^\left\{-2\alpha\right\}\sum_\left\{n=0\right\} \left(-1\right)^n \left\{2\alpha+1 \choose \alpha-n\right\} U_\left\{2n\right\}\left(x\right).\end\left\{align\right\}

### Partial sums

The partial sums of

$f\left(x\right) = \sum_\left\{n = 0\right\}^\infty a_n T_n\left(x\right)$

are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients an are through the use of the inner product as in Galerkin's method and through the use of collocation which is related to interpolation.

As an interpolant, the N coefficients of the (N − 1)th partial sum are usually obtained on the Chebyshev–Gauss–Lobatto[5] points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:

### Polynomial in Chebyshev form

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind. Such a polynomial p(x) is of the form

$p\left(x\right) = \sum_\left\{n=0\right\}^\left\{N\right\} a_n T_n\left(x\right).$

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.

The spread polynomials are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in rational trigonometry.