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 are easily defined recursively, like Fibonacci or Lucas numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Pafnuty Lvovich Chebyshev (Пафну́тий Льво́вич Чебышёв ( –) was a Russian Mathematician. In Mathematics, a polynomial sequence is a Sequence of Polynomials indexed by the nonnegative integers 0 1 2 3. In Mathematics, an orthogonal polynomial sequence is an infinite sequence of real Polynomials p_0\ p_1\ p_2\ \ldots De Moivre's formula, named after Abraham de Moivre, states that for any Complex number (and in particular for any Real number) x and any In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci The Lucas numbers are an Integer sequence named after the mathematician François Édouard Anatole Lucas (1842&ndash1891 who studied both that sequence and the 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 Tchebyshef or Tschebyscheff. Transliteration is the practice of Transcribing a Word or text written in one Writing system into another writing system or system of rules for such practice

The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a polynomial sequence is a Sequence of Polynomials indexed by the nonnegative integers 0 1 2 3.

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. In Mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with Quantitatively In Numerical analysis, Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind. In the mathematical subfield of Numerical analysis, polynomial interpolation is the Interpolation of a given Data set by a Polynomial 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. In the mathematical field of Numerical analysis, Runge's phenomenon is a problem that occurs when using Polynomial interpolation with polynomials of In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematical analysis, the uniform norm assigns to real- or complex -valued bounded functions f the nonnegative number This approximation leads directly to the method of Clenshaw–Curtis quadrature. Clenshaw–Curtis quadrature and Fejér quadrature are methods for Numerical integration, or "quadrature" that are based on an expansion of the integrand

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

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

and

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

for the polynomials of the first and second kind, respectively. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Chebyshev's equation is the second order linear differential equation (1-x^2 {d^2 y \over d x^2} - x {d y \over d x} + p^2 y = 0 where These equations are special cases of the Sturm-Liouville differential equation. In Mathematics and its applications a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855 and Joseph Liouville

## Definition

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

$T_0(x) = 1 \,\!$
$T_1(x) = x \,\!$
$T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x). \,\!$

One example of a generating function for Tn is

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

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

$U_0(x) = 1 \,\!$
$U_1(x) = 2x \,\!$
$U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x). \,\!$

One example of a generating function for Un is

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

### Trigonometric definition

The Chebyshev polynomials of the first kind can be defined by the trigonometric identity:

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

whence:

$T_n(\cos(\theta))=\cos(n\theta) \,\!$

for n = 0, 1, 2, 3, . "Difference equation" redirects here It should not be confused with a Differential equation. In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n "Difference equation" redirects here It should not be confused with a Differential equation. In Mathematics a generating function is a Formal power series whose coefficients encode information about a Sequence a n In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables . . , while the polynomials of the second kind satisfy:

$U_n(\cos(\theta)) = \frac{\sin((n+1)\theta)}{\sin\theta} \,\!$

which is structurally quite similar to the Dirichlet kernel. In Mathematical analysis, the Dirichlet kernel is the collection of functions D_n(x=\sum_{k=-n}^n e^{ikx}=1+2\sum_{k=1}^n\cos(kx=\frac{\sin\left(\left(n

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 via the identity cos2(x) + sin2(x) = 1. De Moivre's formula, named after Abraham de Moivre, states that for any Complex number (and in particular for any Real number) x and any

This identity is extremely 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:

$T_0(x)=\cos\ 0x\ =1 \,\!$

and:

$T_1(\cos(x))=\cos\ (x) \,\!$

one can straightforwardly determine that:

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

and so forth. To trivially check whether the results seem reasonable, sum the coefficients on both sides of the equals sign (that is, setting theta equal to zero, for which the cosine is unity), and one sees that 1 = 2 − 1 in the former expression and 1 = 4 − 3 in the latter.

An immediate corollary is the composition identity (or the "nesting property")

$T_n(T_m(x)) = T_{n\cdot m}(x).\,\!$

### Pell equation definition

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

$T_i^2 - (x^2-1) U_{i-1}^2 = 1 \,\!$

in a ring R[x]. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x [2] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

$T_i + U_{i-1} \sqrt{x^2-1} = (x + \sqrt{x^2-1})^i. \,\!$

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

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

$\frac{d}{dx} \, T_n(x) = n U_{n-1}(x) \mbox{ , } n=1,\ldots$
$T_n(x) = \frac{1}{2} (U_n(x) - \, U_{n-2}(x)).$
$T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x)\,$
$T_n(x) = U_n(x) - x \, U_{n-1}(x).$

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

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

This relationship is used in the Chebyshev spectral method of solving differential equations. Spectral methods are a class of techniques used in Applied mathematics and Scientific computing to numerically solve certain Partial differential equations

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

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

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

\begin{align} 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.

## Explicit formulas

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

$T_n(x) = \begin{cases}\cos(n\arccos(x)), & \ x \in [-1,1] \\\cosh(n \, \mathrm{arccosh}(x)), & \ x \ge 1 \\(-1)^n \cosh(n \, \mathrm{arccosh}(-x)), & \ x \le -1 \\\end{cases} \,\!$
$T_n(x)=\frac{(x+\sqrt{x^2-1})^n+(x-\sqrt{x^2-1})^n}{2} = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} (x^2-1)^k x^{n-2k}$
$U_n(x)=\frac{(x+\sqrt{x^2-1})^{n+1}-(x-\sqrt{x^2-1})^{n+1}}{2\sqrt{x^2-1}} = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (x^2-1)^k x^{n-2k}$
$T_n(x) = 1+n^2 {(x-1)} \prod_{k=1}^{n-1} \left( { 1+{{{x-1}}\over 2 \sin^2\left({k \pi \over n}\right)}}\right)$ (due to M. Hovdan)

## Properties

### Orthogonality

Both the Tn and the Un form a sequence of orthogonal polynomials. In Mathematics, an orthogonal polynomial sequence is an infinite sequence of real Polynomials p_0\ p_1\ p_2\ \ldots The polynomials of the first kind are orthogonal with respect to the weight

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

on the interval [−1,1], i. e. we have:

$\int_{-1}^1 T_n(x)T_m(x)\,\frac{dx}{\sqrt{1-x^2}}=\left\{\begin{matrix}0 &: n\ne m~~~~~\\\pi &: n=m=0\\\pi/2 &: n=m\ne 0\end{matrix}\right. \,\!$

This can be proven by letting x= cos(θ) and using the identity Tn (cos(θ))=cos(nθ). Similarly, the polynomials of the second kind are orthogonal with respect to the weight

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

on the interval [−1,1], i. e. we have:

$\int_{-1}^1 U_n(x)U_m(x)\sqrt{1-x^2}\,dx = \begin{cases}0 &: n\ne m, \\\pi/2 &: n=m.\end{cases} \,\!$

(Note that the weight $\sqrt{1-x^2} \,\!$ is, to within a normalizing constant, the density of the Wigner semicircle distribution). The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the Probability distribution supported on the interval ''R'' the graph of whose

### Minimal ∞-norm

For any given $1 \le n$, among the polynomials of degree n with leading coefficient 1,

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

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

This maximal absolute value is

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

and | f(x) | reaches this maximum exactly n + 1 times: in − 1 and 1 and the other n − 1 extremal points of f.

### 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:

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

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. In Calculus and other branches of Mathematical analysis, an indeterminate form is an Algebraic expression obtained in the context of Limits It can be shown (see proof) that:

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

indeed, the following, more general formula holds:

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

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

Concerning integration, the first derivative of the Tn implies that

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

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

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

### 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 are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. In Numerical analysis, Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind. Using the trigonometric definition and the fact that

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

one can easily prove that the roots of Tn are

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

Similarly, the roots of Un are

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

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. In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. In Differential topology, a critical value of a Differentiable function between Differentiable manifolds is the image of a Critical point In Mathematics, a dessin d'enfant ( French for a "child's drawing" plural dessins d'enfants, "children's drawings" is a type of Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:

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

### 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. In Mathematics, Gegenbauer polynomials or ultraspherical polynomials are a class of Orthogonal polynomials They are named for Leopold Gegenbauer In Mathematics, Jacobi polynomials are a class of Orthogonal polynomials.

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 Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

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 to equal to n. In Mathematics, a Lissajous curve ( Lissajous figure or Bowditch curve) is the graph of the system of Parametric equations

## Examples

This image shows the first few Chebyshev polynomials of the first kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat T0, and T1, T2, T3, T4 and T5.

The first few Chebyshev polynomials of the first kind are

$T_0(x) = 1 \,$
$T_1(x) = x \,$
$T_2(x) = 2x^2 - 1 \,$
$T_3(x) = 4x^3 - 3x \,$
$T_4(x) = 8x^4 - 8x^2 + 1 \,$
$T_5(x) = 16x^5 - 20x^3 + 5x \,$
$T_6(x) = 32x^6 - 48x^4 + 18x^2 - 1 \,$
$T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x \,$
$T_8(x) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \,$
$T_9(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \,$
This image shows the first few Chebyshev polynomials of the second kind in the domain −1¼ < x < 1¼, −1¼ < y < 1¼; the flat U0, and U1, U2, U3, U4 and U5. Although not visible in the image, Un(1) = n + 1 and Un(−1) = (n + 1)(−1)n.

The first few Chebyshev polynomials of the second kind are

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

## As a basis set

The non-smooth function (top) y = − x3H( − x), where H is the Heaviside step function, and (bottom) the 5th partial sum of its Chebyshev expansion. The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative The 7th sum is indistinguishable from the original function at the resolution of the graph.

In the appropriate Sobolev space, the set of Chebyshev polynomials form a complete basis set, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:[3]

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

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. In Mathematics, a Sobolev space is a Vector space of functions equipped with a norm that is a combination of ''Lp'' norms of the function In general an object is complete if nothing needs to be added to it In Mathematics, two Vectors are orthogonal if they are Perpendicular, i In Mathematics, an inner product space is a Vector space with the additional Structure of 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. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions [3] These attributes include:

• The Chebyshev polynomials form a complete orthogonal system. In general an object is complete if nothing needs to be added to it
• The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. In Mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of the Independent In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The smoothness requirement can be relaxed in most cases — as long as there are a finite number of discontinuities in f(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[3], often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem). In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) Spectral methods are a class of techniques used in Applied mathematics and Scientific computing to numerically solve certain Partial differential equations In Mathematics, the Gibbs phenomenon (also known as ringing artifacts) named after the American physicist J

### Partial sums

The partial sums of

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

are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). In Mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with Quantitatively A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Spectral methods are a class of techniques used in Applied mathematics and Scientific computing to numerically solve certain Partial differential equations 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. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, in the area of Numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a Differential In mathematics a collocation method is a method for the numerical solution of Ordinary differential equation and Partial differential equations and In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of

As an interpolant, the N coefficients of the (N − 1)th partial sum are usually obtained on the Chebyshev-Gauss-Lobatto[4] points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. In the mathematical field of Numerical analysis, Runge's phenomenon is a problem that occurs when using Polynomial interpolation with polynomials of This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:

$x_i = -\cos\left(\frac{i \pi}{N - 1}\right) ; \qquad \ i = 0, 1, \dots, N - 1.$

### 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

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

Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm. In the mathematical subfield of Numerical analysis the Clenshaw algorithm (Invented by Charles William Clenshaw) is a recursive method to evaluate