View source for Chebyshev polynomials

{{Short description|Polynomial sequence}}
{{distinguish|discrete Chebyshev polynomials}}
{{Use American English|date = March 2019}}
{{use dmy dates|date=August 2020}}
[[File:Chebyshev Polynomials of the First Kind.svg|thumb|Plot of the first five {{mvar|T<sub>n</sub>}} Chebyshev polynomials (first kind)]]
[[File:Chebyshev Polynomials of the Second Kind.svg|thumb|Plot of the first five {{mvar|U<sub>n</sub>}} Chebyshev polynomials (second kind)]]

The '''Chebyshev polynomials''' are two sequences of [[orthogonal polynomials]] related to the [[trigonometric functions|cosine and sine functions]], notated as <math>T_n(x)</math> and <math>U_n(x)</math>. They can be defined in several equivalent ways, one of which starts with [[trigonometric functions]]:

The '''Chebyshev polynomials of the first kind''' <math>T_n</math> are defined by

<math display="block">T_n(\cos \theta) = \cos(n\theta).</math>

Similarly, the '''Chebyshev polynomials of the second kind''' <math>U_n</math> are defined by

<math display="block">U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big).</math>

That these expressions define polynomials in <math>\cos\theta</math> is not obvious at first sight but can be shown using [[de Moivre's formula]] (see [[#Trigonometric definition|below]]).

The Chebyshev polynomials {{math| ''T<sub>n</sub>''}} are polynomials with the largest possible leading coefficient whose [[absolute value]] on the [[interval (mathematics)|interval]] {{closed-closed|−1, 1}} is bounded by 1. They are also the "extremal" polynomials for many other properties.<ref>{{cite book |last=Rivlin |first=Theodore J. |author-link=Theodore J. Rivlin |year=1974 |title=The Chebyshev Polynomials |edition=1st |chapter=Chapter &nbsp;2, Extremal properties |pages=56–123 |series=Pure and Applied Mathematics |publisher=Wiley-Interscience [John Wiley & Sons] |place=New York-London-Sydney |isbn=978-047172470-4}}</ref>

In 1952, [[Cornelius Lanczos]] showed that the Chebyshev polynomials are important in [[approximation theory]] for the solution of linear systems;<ref>{{ cite journal  | title=Solution of systems of linear equations by minimized iterations | year=1952 | pages=33 | journal=Journal of Research of the National Bureau of Standards | doi=10.6028/jres.049.006 | volume=49 | issue=1 | last1=Lanczos | first1= C. | doi-access=free }}</ref>  the [[root of a polynomial|roots]] of {{math|''T<sub>n</sub>''(''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]].<ref>Chebyshev first presented his eponymous polynomials in a paper read before the St. Petersburg Academy in 1853: {{pb}} {{cite journal |last=Chebyshev |first=P. L. |year=1854 |title=Théorie des mécanismes connus sous le nom de parallélogrammes |language=fr |journal=Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg |volume=7 |pages=539–586 |url=https://archive.org/details/mmoiresprsentsla07impe/page/537/ }} Also published separately as {{cite book |last = Chebyshev |first= P. L. |year=1853 |title=Théorie des mécanismes connus sous le nom de parallélogrammes |place=St. Petersburg |publisher=Imprimerie de l'Académie Impériale des Sciences |doi=10.3931/E-RARA-120037 |doi-access=free }}</ref> The letter {{mvar|T}} is used because of the alternative [[transliteration]]s of the name ''Chebyshev'' as {{lang|fr|Tchebycheff}}, {{lang|fr|Tchebyshev}} (French) or {{lang|de|Tschebyschow}} (German).

==Definitions==

=== Recurrence definition ===

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

<math display="block">\begin{align}
 T_0(x) & = 1, \\
 T_1(x) & = x, \\
 T_{n+1}(x) & = 2 x\,T_n(x) - T_{n-1}(x).
\end{align}</math>

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

<math display="block">\begin{align}
 U_0(x) & = 1, \\
 U_1(x) & = 2 x, \\
 U_{n+1}(x) & = 2 x\,U_n(x) - U_{n-1}(x),
\end{align}</math>
which differs from the above only by the rule for ''n=1''.

===Trigonometric definition===

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

<math display="block">T_n(\cos\theta) = \cos(n\theta)</math>

and

<math display="block">U_n(\cos\theta) = \frac{\sin\big((n + 1)\theta\big)}{\sin\theta},</math>

for {{math|1=''n'' = 0, 1, 2, 3, …}}.

An equivalent way to state this is via exponentiation of a [[complex number]]: given a complex number {{math|1=''z'' = ''a'' + ''bi''}} with absolute value of one,

<math display="block">z^n = T_n(a) + ib U_{n-1}(a).</math>

Chebyshev polynomials can be defined in this form when studying [[trigonometric polynomials]].<ref>{{Cite journal |last=Schaeffer |first=A. C. |date=1941 |title=Inequalities of A. Markoff and S. Bernstein for polynomials and related functions |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-47/issue-8/Inequalities-of-A-Markoff-and-S-Bernstein-for-polynomials-and/bams/1183503783.full |journal=Bulletin of the American Mathematical Society |volume=47 |issue=8 |pages=565–579 |doi=10.1090/S0002-9904-1941-07510-5 |issn=0002-9904|doi-access=free }}</ref>

That <math>\cos(nx)</math> is an <math>n</math>th-[[degree of a polynomial|degree]] polynomial in <math>\cos(x)</math> can be seen by observing that <math>\cos(nx)</math> is the [[complex number|real part]] of one side of [[de Moivre's formula]]:

<math display="block">\cos n \theta + i \sin n \theta = (\cos \theta + i \sin \theta)^n.</math>

The real part of the other side is a polynomial in <math>\cos(x)</math> and <math>\sin(x)</math>, in which all powers of <math>\sin(x)</math> are [[parity (mathematics)|even]] and thus replaceable through the identity <math>\cos^2(x)+\sin^2(x)=1</math>. By the same reasoning, <math>\sin(nx)</math> is the [[complex number|imaginary part]] of the polynomial, in which all powers of <math>\sin(x)</math> are [[parity (mathematics)|odd]] and thus, if one factor of <math>\sin(x)</math> is factored out, the remaining factors can be replaced to create a <math>n-1</math>st-degree polynomial in <math>\cos(x)</math>.

For <math>x</math> outside the interval [-1,1], the above definition implies

<math display="block">T_n(x) = \begin{cases}
 \cos(n \arccos x)                          & \text{ if }~ |x| \le 1, \\
 \cosh(n \operatorname{arcosh} x)           & \text{ if }~ x \ge 1, \\ 
 (-1)^n \cosh(n \operatorname{arcosh}(-x) ) & \text{ if }~ x \le -1.
\end{cases}</math>

===Commuting polynomials definition===

Chebyshev polynomials can also be characterized by the following theorem:<ref>{{cite journal|first=J. F. |last=Ritt |author-link=Joseph Ritt |doi=10.1090/S0002-9947-1922-1501189-9 |title=Prime and Composite Polynomials |journal=Trans. Amer. Math. Soc. |year=1922|volume=23 |pages=51–66 | url=https://www.ams.org/journals/tran/1922-023-01/S0002-9947-1922-1501189-9 |doi-access=free|url-access=subscription }}</ref>

If <math> F_n(x)</math> is a family of monic polynomials with coefficients in a field of characteristic <math>0</math> such that <math> \deg F_n(x) = n</math> and <math> F_m(F_n(x)) = F_n(F_m(x))</math> for all 
<math>m</math> and <math> n</math>, then, up to a simple change of variables, either <math> F_n(x) = x^n</math> for all <math> n</math> or 
<math>F_n(x) = 2\cdot T_n(x/2)</math> for all <math> n</math>.

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

<math display="block">T_n(x)^2 - \left(x^2 - 1\right) U_{n-1}(x)^2 = 1</math>

in a [[ring (mathematics)|ring]] <math>R[x]</math>.<ref>{{cite thesis |first=Jeroen |last=Demeyer |url=http://cage.ugent.be/~jdemeyer/phd.pdf |title=Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields |archive-url=https://web.archive.org/web/20070702185523/https://cage.ugent.be/~jdemeyer/phd.pdf |archive-date=2007-07-02 |degree=Ph.D. |year=2007 |page=70}}</ref> Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

<math display="block">T_n(x) + U_{n-1}(x)\,\sqrt{x^2-1} = \left(x + \sqrt{x^2-1}\right)^n~. </math>

===Generating functions===

The [[generating function|ordinary generating function]] for <math>T_n</math> is

<math display="block">\sum_{n=0}^\infty T_n(x)\,t^n = \frac{1 - tx}{1 - 2tx + t^2}.</math>

There are several other [[generating function]]s for the Chebyshev polynomials; the [[exponential generating function]] is

<math display="block">\begin{align}
\sum_{n=0}^\infty T_n(x) \frac{t^n}{n!}
  &= {\tfrac{1}{2}} \Bigl({\exp}\Bigl({\textstyle t\bigl(x - \sqrt{x^2 - 1}~\!\bigr)}\Bigr)
                      + {\exp}\Bigl({\textstyle t\bigl(x + \sqrt{x^2 - 1}~\!\bigr)}\Bigr)\Bigr) \\
  &= e^{tx} \cosh\left({\textstyle t\sqrt{x^2 - 1} }~\! \right).
\end{align}</math>

The generating function relevant for 2-dimensional [[potential theory]] and [[Cylindrical multipole moments|multipole expansion]] is

<math display="block">\sum\limits_{n=1}^\infty T_{n}(x)\,\frac{t^n}{n} = \ln\left(\frac{1}{\sqrt{1 - 2tx + t^2 }}\right).</math>

The ordinary generating function for {{mvar|U<sub>n</sub>}} is

<math display="block">\sum_{n=0}^\infty U_n(x)\,t^n = \frac{1}{1 - 2tx + t^2},</math>

and the exponential generating function is

<math display="block">
 \sum_{n=0}^\infty U_n(x) \frac{t^n}{n!} = e^{tx} \biggl(\!\cosh\left(t\sqrt{x^2 - 1}\right) + \frac{x}{\sqrt{x^2 - 1}} \sinh\left(t\sqrt{x^2 - 1}\right)\biggr).
</math>

==Relations between the two kinds of Chebyshev polynomials==
The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of [[Lucas sequence]]s <math>\tilde V_n(P,Q)</math> and <math>\tilde U_n(P,Q)</math> with parameters <math>P=2x</math> and <math>Q=1</math>:

<math display="block">\begin{align}
{\tilde U}_n(2x,1) &= U_{n-1}(x), \\
{\tilde V}_n(2x,1) &= 2\, T_n(x).
\end{align}</math>

It follows that they also satisfy a pair of mutual recurrence equations:{{sfn|Bateman|Bateman Manuscript Project|1953|loc=[https://archive.org/details/highertranscende02bate/page/184/ {{pgs|184}}, eqs. 3–4]}}

<math display="block">\begin{align}
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{align}</math>

The second of these may be rearranged using the [[#Recurrence definition|recurrence definition]] for the Chebyshev polynomials of the second kind to give:

<math display="block">T_n(x) = \frac{1}{2} \big(U_n(x) - U_{n-2}(x)\big).</math>

Using this formula iteratively gives the sum formula:

<math display="block">
U_n(x) = \begin{cases}
2\sum_{\text{ odd }j>0}^n T_j(x)      & \text{ for odd }n.\\
2\sum_{\text{ even }j\ge 0}^n T_j(x) - 1      & \text{ for even }n,
\end{cases}
</math>

while replacing <math>U_n(x)</math> and <math>U_{n-2}(x)</math> using the [[#Differentiation and integration|derivative formula]] for <math>T_n(x)</math> gives the recurrence relationship for the derivative of <math>T_n</math>:

<math display="block">2\,T_n(x) = \frac{1}{n+1}\, \frac{\mathrm{d}}{\mathrm{d}x}\, T_{n+1}(x) - \frac{1}{n-1}\,\frac{\mathrm{d}}{\mathrm{d}x}\, T_{n-1}(x), \qquad n=2,3,\ldots</math>

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

[[Turán's inequalities]] for the Chebyshev polynomials are:<ref>{{citation|mr=0040487 | last1=Beckenbach | first1= E. F.| last2= Seidel|first2= W.| last3= Szász|first3= Otto | title=Recurrent determinants of Legendre and of ultraspherical polynomials | journal=Duke Math. J. | volume= 18 | year=1951 | pages= 1–10 | doi=10.1215/S0012-7094-51-01801-7}}</ref>

<math display="block">\begin{align}
T_n(x)^2 - T_{n-1}(x)\,T_{n+1}(x)&= 1-x^2 > 0 &&\text{ for } -1<x<1 &&\text{ and }\\
U_n(x)^2 - U_{n-1}(x)\,U_{n+1}(x)&= 1 > 0~.
\end{align}</math>

The [[integral]] relations are{{sfn|Bateman|Bateman Manuscript Project|1953|loc=[https://archive.org/details/highertranscende02bate/page/187/ {{pgs| 187}}, eqs. 47–48]}}{{sfn|Mason|Handscomb|2002}}

<math display="block">\begin{align}
\int_{-1}^1 \frac{T_n(y)}{y-x} \, \frac{\mathrm{d}y}{\sqrt{1 - y^2}} &= \pi\,U_{n-1}(x)~, \\[1.5ex]
\int_{-1}^1\frac{U_{n-1}(y)}{y-x}\, \sqrt{1 - y^2}\mathrm{d}y &= -\pi\,T_n(x)
\end{align}</math>

where integrals are considered as principal value.

==Explicit expressions==

Using the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expressions, valid for any real {{tmath|x}}:{{cn|date=June 2025}}

<math display="block">\begin{align}
T_n(x)
&= \tfrac{1}{2} \Big( \bigl({\textstyle x-\sqrt{x^2-1}\!~}\bigr)^n + \bigl({\textstyle x+\sqrt{x^2-1}\!~}\bigr)^n \Big) \\[5mu]
&= \tfrac{1}{2} \Big( \bigl({\textstyle x-\sqrt{x^2-1}\!~}\bigr)^n + \bigl({\textstyle x-\sqrt{x^2-1}\!~}\bigr)^{-n} \Big).
\end{align}</math>

The two are equivalent because <math>\textstyle \bigl(x + \sqrt{x^2 - 1}\!~\bigr)\bigl(x - \sqrt{x^2 - 1}\!~\bigr) = 1</math>.

An explicit form of the Chebyshev polynomial in terms of monomials <math>x^k</math> follows from [[de Moivre's formula]]:

<math display="block">T_n(\cos(\theta)) = \operatorname{Re}(\cos n \theta + i \sin n \theta) = \operatorname{Re}((\cos \theta + i \sin \theta)^n),</math>

where <math>\mathrm{Re}</math> denotes the [[Complex number#Notation|real part]] of a complex number. Expanding the formula, one gets

<math display="block">(\cos \theta + i \sin \theta)^n = \sum\limits_{j=0}^n \binom{n}{j} i^j \sin^j \theta \cos^{n-j} \theta.</math>

The real part of the expression is obtained from summands corresponding to even indices. Noting <math>i^{2j} = (-1)^j</math> and <math>\sin^{2j} \theta = (1-\cos^2 \theta)^j</math>, one gets the explicit formula:

<math display="block">\cos n \theta = \sum\limits_{j=0}^{\lfloor n / 2 \rfloor} \binom{n}{2j} (\cos^2 \theta - 1)^j \cos^{n-2j} \theta,</math>

which in turn means that

<math display="block">T_n(x) = \sum\limits_{j=0}^{\lfloor n / 2 \rfloor} \binom{n}{2j} (x^2-1)^j x^{n-2j}.</math>

This can be written as a {{math|<sub>2</sub>''F''<sub>1</sub>}} [[hypergeometric function]]:

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

with inverse<ref name=Cody>{{cite journal |first1=W. J. |last1=Cody |title=A survey of practical rational and polynomial approximation of functions |year=1970 |journal=SIAM Review |volume=12 |number=3 |pages=400–423 |doi=10.1137/1012082}}</ref><ref name=Mathar>{{cite journal
 | last=Mathar | first=Richard J. 
 | year=2006 
 | title=Chebyshev series expansion of inverse polynomials
 | journal=Journal of Computational and Applied Mathematics
 | volume=196 | issue=2 | pages=596–607
 | doi=10.1016/j.cam.2005.10.013 | doi-access=free
 | arxiv=math/0403344
 }}</ref>

<math display="block">x^n = 2^{1-n}\mathop{{\sum}'}^n_{j=0\atop j \equiv n \pmod 2} \!\!\binom{n}{\tfrac{n-j}{2}}\!\;T_j(x),</math>

where the prime at the summation symbol indicates that the contribution of <math>j=0</math> needs to be halved if it appears.

A related expression for <math>T_n</math> as a sum of monomials with binomial coefficients and powers of two is

<math display="block">
 T_n(x) = \sum\limits_{m=0}^{\left\lfloor \frac{n}{2} \right\rfloor} (-1)^m \left(\binom{n - m}{m} + \binom{n - m - 1}{n - 2m}\right) \cdot 2^{n-2m-1} \cdot x^{n-2m}.</math>

Similarly, <math>U_n</math> can be expressed in terms of hypergeometric functions:

<math display="block">\begin{align}
U_n(x) &= \frac{\left (x+\sqrt{x^2-1} \right )^{n+1} - \left (x-\sqrt{x^2-1} \right )^{n+1}}{2\sqrt{x^2-1}} \\
 &= \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{n+1}{2k+1} \left (x^2-1 \right )^k x^{n-2k} \\
 &= x^n \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{n+1}{2k+1} \left (1 - x^{-2} \right )^k \\
 &= \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} \binom{2k-(n+1)}{k}~(2x)^{n-2k} & \text{ for } n > 0 \\
 &= \sum_{k=0}^{\left \lfloor {n}/{2} \right \rfloor} (-1)^k \binom{n-k}{k}~(2x)^{n-2k} & \text{ for } n > 0 \\
 &= \sum_{k=0}^{n}(-2)^{k} \frac{(n+k+1)!} {(n-k)!(2k+1)!}(1 - x)^k & \text{ for } n > 0 \\
 &= (n + 1)\, {}_2F_1\big(-n, n + 2; \tfrac{3}{2}; \tfrac{1}{2}(1 - x)\big).
\end{align}</math>

==Properties==

===Symmetry===

<math display="block">\begin{align}
 T_n(-x) &= (-1)^n\, T_n(x),\\[1ex]
 U_n(-x) &= (-1)^n\, U_n(x).
\end{align}</math>

That is, Chebyshev polynomials of even order have [[even and odd functions|even symmetry]] and therefore contain only even powers of <math>x</math>. Chebyshev polynomials of odd order have [[even and odd functions|odd symmetry]] and therefore contain only odd powers of <math>x</math>.

===Roots and extrema===
A Chebyshev polynomial of either kind with degree {{mvar|n}} has {{mvar|n}} different [[simple root]]s, called '''Chebyshev roots''', in the interval {{closed-closed|−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:

<math display="block">\cos\left((2k+1)\frac{\pi}{2}\right)=0</math>

one can show that the roots of <math>T_n</math> are:

<math display="block"> x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1.</math>

Similarly, the roots of <math>U_n</math> are:

<math display="block"> x_k = \cos\left(\frac{k}{n+1}\pi\right),\quad k=1,\ldots,n.</math>

The [[Maxima and minima|extrema]] of <math>T_n</math> on the interval <math>-1\leq x\leq 1</math> are located at:

<math display="block"> x_k = \cos\left(\frac{k}{n}\pi\right),\quad k=0,\ldots,n.</math>

One unique property of the Chebyshev polynomials of the first kind is that on the interval <math>-1\leq x\leq 1</math> all of the [[Maxima and minima|extrema]] have values that are either −1 or 1. Thus these polynomials have only two finite [[Critical value (critical point)|critical value]]s, the defining property of [[Shabat polynomial]]s. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:

<math display="block">\begin{align}
T_n(1) &= 1 \\
T_n(-1) &= (-1)^n \\
U_n(1) &= n+1 \\
U_n(-1) &= (-1)^n (n+1).
\end{align}</math>

The [[Maxima and minima|extrema]] of <math>T_n(x)</math> on the interval <math>-1 \leq x \leq 1</math> where <math>n>0</math> are located at <math>n+1</math> values of <math>x</math>. They are <math> \pm 1</math>, or <math>  \cos\left(\frac{2\pi k}{d}\right)</math> where <math>d > 2</math>, <math>d \;|\; 2n</math>, <math>0 < k < d/2</math> and <math>(k, d) = 1</math>, i.e., <math>k</math> and <math>d</math> are relatively prime numbers.

Specifically ([[Minimal polynomial of 2cos(2pi/n)]]<ref name=Gurtas>{{cite journal |first1=Y. Z. |last1=Gürtaş |title=Chebyshev Polynomials and the minimal polynomial of <math>\cos (2 \pi/n)</math> |year=2017 |journal=American Mathematical Monthly |volume=124 |number=1 |pages=74–78 |doi=10.4169/amer.math.monthly.124.1.74|s2cid=125797961 }}</ref><ref name=Wolfram0>{{cite journal |first1=D. A. |last1=Wolfram |title=Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of <math>\cos (2 \pi /d )</math> |year=2022 |journal=American Mathematical Monthly |volume=129 |number=2 |pages=172–176 |doi=10.1080/00029890.2022.2005391|s2cid=245808448 }}</ref>) when <math>n</math> is even: 
* <math>T_n(x) = 1</math> if <math>x = \pm 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>n/2 + 1</math> such values of <math>x</math>.
* <math>T_n(x) = -1</math> if  <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>n/2</math> such values of <math>x</math>.

When <math>n</math> is odd: 
* <math>T_n(x) = 1</math> if <math>x = 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>(n+1)/2</math> such values of <math>x</math>.
* <math>T_n(x) = -1</math> if  <math>x = -1</math>, or <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>(n+1)/2</math> such values of <math>x</math>.

===Differentiation and integration===
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it can be shown that:

<math display="block">\begin{align}
\frac{\mathrm{d}T_n}{\mathrm{d}x} &= n U_{n - 1} \\
\frac{\mathrm{d}U_n}{\mathrm{d}x} &= \frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1} \\
\frac{\mathrm{d}^2 T_n}{\mathrm{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}.
\end{align}</math>

The last two formulas can be numerically troublesome due to the division by zero ({{Sfrac|0|0}} [[indeterminate form]], specifically) at <math>x=1</math> and <math>x=-1</math>. By [[L'Hôpital's rule]]:

<math display="block">\begin{align}
\left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = 1} \!\! &= \frac{n^4 - n^2}{3}, \\
\left. \frac{\mathrm{d}^2 T_n}{\mathrm{d}x^2} \right|_{x = -1} \!\! &= (-1)^n \frac{n^4 - n^2}{3}.
\end{align}</math>

More generally,

<math display="block">\left.\frac{\mathrm{d}^p T_n}{\mathrm{d}x^p} \right|_{x = \pm 1} \!\! = (\pm 1)^{n+p}\prod_{k=0}^{p-1}\frac{n^2-k^2}{2k+1}~,</math>

which is of great use in the numerical solution of [[eigenvalue]] problems.

Also, we have:

<math display="block">\frac{\mathrm{d}^p}{\mathrm{d}x^p}\,T_n(x) = 2^p\,n\mathop{{\sum}'}_{0\leq k\leq n-p\atop k \,\equiv\, n-p \pmod 2}
\binom{\frac{n+p-k}{2}-1}{\frac{n-p-k}{2}}\frac{\left(\frac{n+p+k}{2}-1\right)!}{\left(\frac{n-p+k}{2}\right)!}\,T_k(x),~\qquad p \ge 1,</math>

where the prime at the summation symbols means that the term contributed by {{math|1=''k'' = 0}} is to be halved, if it appears.

Concerning integration, the first derivative of the {{mvar|T<sub>n</sub>}} implies that:

<math display="block">\int U_n\, \mathrm{d}x = \frac{T_{n + 1}}{n + 1}</math>

and the recurrence relation for the first kind polynomials involving derivatives establishes that for <math>n\geq 2</math>:

<math display="block">\int T_n\, \mathrm{d}x = \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}.</math>

The last formula can be further manipulated to express the integral of <math>T_n</math> as a function of Chebyshev polynomials of the first kind only:

<math display="block">\begin{align}
\int T_n\, \mathrm{d}x &= \frac{n}{n^2 - 1} T_{n + 1} - \frac{1}{n - 1} T_1 T_n \\
&= \frac{n}{n^2 - 1}\,T_{n + 1} - \frac{1}{2(n - 1)}\,(T_{n + 1} + T_{n - 1}) \\
&= \frac{1}{2(n + 1)}\,T_{n + 1} - \frac{1}{2(n - 1)}\,T_{n - 1}.
\end{align}</math>

Furthermore, we have:

<math display="block">\int_{-1}^1 T_n(x)\, \mathrm{d}x =
\begin{cases}
\frac{(-1)^n + 1}{1 - n^2} & \text{ if }~ n \ne 1 \\
0                          & \text{ if }~ n = 1.
\end{cases}</math>

===Products of Chebyshev polynomials===
The Chebyshev polynomials of the first kind satisfy the relation:

<math display="block">T_m(x)\,T_n(x) = \tfrac{1}{2}\!\left(T_{m+n}(x) + T_{|m-n|}(x)\right)\!,\qquad \forall m,n \ge 0,</math>

which is easily proved from the [[List of trigonometric identities#Product-to-sum and sum-to-product identities|product-to-sum formula]] for the cosine:

<math display="block">2 \cos \alpha \, \cos \beta = \cos (\alpha + \beta) + \cos (\alpha - \beta).</math>

For <math>n=1</math> this results in the already known recurrence formula, just arranged differently, and with <math>n=2</math> it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest {{mvar|m}}) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:

<math display="block">\begin{align}
  T_{2n}(x) &= 2\,T_n^2(x)           - T_0(x) &&= 2 T_n^2(x) - 1, \\
T_{2n+1}(x) &= 2\,T_{n+1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n+1}(x)\,T_n(x) - x, \\
T_{2n-1}(x) &= 2\,T_{n-1}(x)\,T_n(x) - T_1(x) &&= 2\,T_{n-1}(x)\,T_n(x) - x .
\end{align}</math>

The polynomials of the second kind satisfy the similar relation:

<math display="block"> T_m(x)\,U_n(x) = \begin{cases}
\frac{1}{2}\left(U_{m+n}(x) + U_{n-m}(x)\right),  & ~\text{ if }~ n \ge m-1,\\
\\
\frac{1}{2}\left(U_{m+n}(x) - U_{m-n-2}(x)\right), & ~\text{ if }~ n \le m-2.
\end{cases} </math>

(with the definition <math>U_{-1}\equiv 0</math> by convention ). They also satisfy:

<math display="block"> U_m(x)\,U_n(x) = \sum_{k=0}^n\,U_{m-n+2k}(x) = \sum_\underset{\text{ step 2 }}{p=m-n}^{m+n} U_p(x)~.</math>

for <math>m\geq n</math>.
For <math>n=2</math> this recurrence reduces to:

<math display="block"> U_{m+2}(x) = U_2(x)\,U_m(x) - U_m(x) - U_{m-2}(x) = U_m(x)\,\big(U_2(x) - 1\big) - U_{m-2}(x)~,</math>

which establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether <math>m</math> starts with 2 or 3.

===Composition and divisibility properties===
The trigonometric definitions of <math>T_n</math> and <math>U_n</math> imply the composition or nesting properties:<ref>{{citation|last1=Rayes|first1=M. O.|last2=Trevisan|first2=V.|last3=Wang|first3=P. S.|title=Factorization properties of chebyshev polynomials|journal=Computers & Mathematics with Applications|volume=50|issue=8–9|year=2005|pages=1231–1240|doi=10.1016/j.camwa.2005.07.003|doi-access=free}}</ref>

<math display="block">\begin{align}
T_{mn}(x) &= T_m(T_n(x)),\\
U_{mn-1}(x) &= U_{m-1}(T_n(x))U_{n-1}(x).
\end{align}
</math>

For <math>T_{mn}</math> the order of composition may be reversed, making the family of polynomial functions <math>T_n</math> a [[commutative]] [[semigroup]] under composition.

Since <math>T_m(x)</math> is divisible by <math>x</math> if <math>m</math> is odd, it follows that <math>T_{mn}(x)</math> is divisible by <math>T_n(x)</math> if <math>m</math> is odd. Furthermore, <math>U_{mn-1}(x)</math> is divisible by <math>U_{n-1}(x)</math>, and in the case that <math>m</math> is even, divisible by <math>T_n(x)U_{n-1}(x)</math>.

===Orthogonality===
Both <math>T_n</math> and <math>U_n</math> form a sequence of [[orthogonal polynomials]]. The polynomials of the first kind <math>T_n</math> are orthogonal with respect to the weight:

<math display="block">\frac{1}{\sqrt{1 - x^2}},</math>

on the interval {{closed-closed|−1, 1}}, i.e. we have:

<math display="block">\int_{-1}^1 T_n(x)\,T_m(x)\,\frac{\mathrm{d}x}{\sqrt{1-x^2}} = 
\begin{cases}
0             & ~\text{ if }~ n \ne m, \\[5mu]
\pi           & ~\text{ if }~ n=m=0, \\[5mu]
\frac{\pi}{2} & ~\text{ if }~ n=m \ne 0.
\end{cases}</math>

This can be proven by letting <math>x=\cos(\theta)</math> and using the defining identity <math>T_n(\cos(\theta)=\cos(n\theta)</math>.

Similarly, the polynomials of the second kind {{mvar|U<sub>n</sub>}} are orthogonal with respect to the weight:

<math display="block">\sqrt{1-x^2}</math>
on the interval {{closed-closed|−1, 1}}, i.e. we have:

<math display="block">\int_{-1}^1 U_n(x)\,U_m(x)\,\sqrt{1-x^2} \,\mathrm{d}x =
\begin{cases}
0             & ~\text{ if }~ n \ne m, \\[5mu]
\frac{\pi}{2} & ~\text{ if }~ n = m.
\end{cases}</math>

(The measure <math>\sqrt{1-x^2}\, dx</math> is, to within a normalizing constant, the [[Wigner semicircle distribution]].)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the [[Chebyshev equation|Chebyshev differential equations]]:

<math display="block">\begin{align}
(1 - x^2)T_n'' - xT_n' + n^2 T_n &= 0, \\[1ex]
(1 - x^2)U_n'' - 3xU_n' + n(n + 2) U_n &= 0,
\end{align}</math>
which are [[Sturm–Liouville problem|Sturm–Liouville differential equations]]. It is a general feature of such [[differential equation]]s that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to [[Sturm–Liouville problem|those equations]].)

The <math>T_n</math> also satisfy a discrete orthogonality condition:

<math display="block">\sum_{k=0}^{N-1}{T_i(x_k)\,T_j(x_k)} =
\begin{cases}
0           & ~\text{ if }~ i \ne j, \\[5mu]
N           & ~\text{ if }~ i = j = 0, \\[5mu]
\frac{N}{2} & ~\text{ if }~ i = j \ne 0,
\end{cases} </math>

where <math>N</math> is any integer greater than <math>\max(i,j)</math>,{{sfn|Mason|Handscomb|2002}} and the <math>x_k</math> are the <math>N</math> [[Chebyshev nodes]] (see above) of <math>T_N(x)</math>:

<math display="block">x_k = \cos\left(\pi\,\frac{2k+1}{2N}\right) \quad ~\text{ for }~ k = 0, 1, \dots, N-1.</math>

For the polynomials of the second kind and any integer <math>N>i+j</math> with the same Chebyshev nodes <math>x_k</math>, there are similar sums:

<math display="block">\sum_{k=0}^{N-1}{U_i(x_k)\,U_j(x_k)\left(1-x_k^2\right)} =
\begin{cases}
0           & \text{ if }~ i \ne j, \\[5mu]
\frac{N}{2} & \text{ if }~ i = j,
\end{cases}</math>

and without the weight function:

<math display="block">\sum_{k=0}^{N-1}{ U_i(x_k) \, U_j(x_k) } =
\begin{cases}
0                         & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu]
N \cdot (1 + \min\{i,j\}) & ~\text{ if }~ i \equiv j\pmod{2}.
\end{cases} </math>

For any integer <math>N>i+j</math>, based on the <math>N</math>} zeros of <math>U_N(x)</math>:

<math display="block">y_k = \cos\left(\pi\,\frac{k+1}{N+1}\right) \quad ~\text{ for }~ k=0, 1, \dots, N-1,</math>

one can get the sum:

<math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)(1-y_k^2)} =
\begin{cases}
0 & ~\text{ if } i \ne j, \\[5mu]
\frac{N+1}{2} & ~\text{ if } i = j,
\end{cases}</math>

and again without the weight function:

<math display="block">\sum_{k=0}^{N-1}{U_i(y_k)\,U_j(y_k)} =
\begin{cases}
0  & ~\text{ if }~ i \not\equiv j \pmod{2}, \\[5mu]
\bigl(\min\{i,j\} + 1\bigr)\bigl(N-\max\{i,j\}\bigr)  & ~\text{ if }~ i \equiv j\pmod{2}.
\end{cases}</math>

===Minimal {{math|∞}}-norm===
For any given <math>n\geq 1</math>, among the polynomials of degree <math>n</math> with leading coefficient 1 ([[monic polynomial|monic]] polynomials):

<math display="block">f(x) = \frac{1}{\,2^{n-1}\,}\,T_n(x)</math>

is the one of which the maximal absolute value on the interval {{closed-closed|−1, 1}} is minimal.

This maximal absolute value is:

<math display="block">\frac1{2^{n-1}}</math>

and <math>|f(x)|</math> reaches this maximum exactly <math>n+1</math> times at:

<math display="block">x = \cos \frac{k\pi}{n}\quad\text{for }0 \le k \le n.</math>

{{Math proof | proof =
Let's assume that <math>w_n(x)</math> is a polynomial of degree <math>n</math> with leading coefficient 1 with maximal absolute value on the interval {{closed-closed|−1, 1}} less than {{math|1&thinsp;/&thinsp;2<sup>''n''&nbsp;−&thinsp;1</sup>}}.

Define

<math display="block">f_n(x) = \frac{1}{\,2^{n-1}\,}\,T_n(x) - w_n(x)</math>

Because at extreme points of {{mvar|T<sub>n</sub>}} we have

<math display="block">\begin{align}
|w_n(x)| &< \left|\frac1{2^{n-1}}T_n(x)\right| \\
f_n(x) &> 0 \qquad \text{ for }~ x = \cos \frac{2k\pi}{n} ~&&\text{ where } 0 \le 2k \le n \\
f_n(x) &< 0 \qquad \text{ for }~ x = \cos \frac{(2k + 1)\pi}{n} ~&&\text{ where } 0 \le 2k + 1 \le n
\end{align}</math>

From the [[intermediate value theorem]], {{math|''f<sub>n</sub>''(''x'')}} has at least {{mvar|n}} roots.  However, this is impossible, as {{math|''f<sub>n</sub>''(''x'')}} is a polynomial of degree {{math|''n'' − 1}}, so the [[fundamental theorem of algebra]] implies it has at most {{math|''n'' − 1}} roots.
}}

====Remark====
By the [[equioscillation theorem]], among all the polynomials of degree {{math|≤&thinsp;''n''}}, the polynomial {{mvar|f}} minimizes {{math|{{norm|&thinsp;''f''&thinsp;}}<sub>∞</sub>}} on {{closed-closed|−1, 1}} [[if and only if]] there are {{math|''n'' + 2}} points {{math|−1 ≤ ''x''<sub>0</sub> < ''x''<sub>1</sub> < ⋯ < ''x''<sub>''n'' + 1</sub> ≤ 1}} such that {{math|1={{abs|&thinsp;''f''(''x<sub>i</sub>'')}} = {{norm|&thinsp;''f''&thinsp;}}<sub>∞</sub>}}.

Of course, the null polynomial on the interval {{closed-closed|−1, 1}} can be approximated by itself and minimizes the {{math|∞}}-norm.

Above, however, {{math|{{abs|&thinsp;''f''&thinsp;}}}} reaches its maximum only {{math|''n'' + 1}} times because we are searching for the best polynomial of degree {{math|''n'' ≥ 1}} (therefore the theorem evoked previously cannot be used).

===Chebyshev polynomials as special cases of more general polynomial families===
The Chebyshev polynomials are a special case of the ultraspherical or [[Gegenbauer polynomials]] <math>C_n^{(\lambda)}(x)</math>, which themselves are a special case of the [[Jacobi polynomials]] <math>P_n^{(\alpha,\beta)}(x)</math>:

<math display="block">\begin{align}
T_n(x) &= \frac{n}{2} \lim_{q \to 0}  \frac{1}{q}\,C_n^{(q)}(x) \qquad ~\text{ if }~ n \ge 1, \\
 &= \frac{1}{\binom{n-\frac{1}{2}}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x) = \frac{2^{2n}}{\binom{2n}{n}} P_n^{\left(-\frac{1}{2}, -\frac{1}{2}\right)}(x)~,
\\[2ex]
U_n(x) & = C_n^{(1)}(x)\\
 &= \frac{n+1}{\binom{n+\frac{1}{2}}{n}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x) = \frac{2^{2n+1}}{\binom{2n+2}{n+1}} P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x)~.
\end{align}</math>

Chebyshev polynomials are also a special case of [[Dickson polynomial]]s:

<math display="block">D_n(2x\alpha,\alpha^2)= 2\alpha^{n}T_n(x) \, </math> 

<math display="block">E_n(2x\alpha,\alpha^2)= \alpha^{n}U_n(x). \, </math>

In particular, when <math>\alpha=\tfrac{1}{2}</math>, they are related by <math>D_n(x,\tfrac{1}{4}) = 2^{1-n}T_n(x)</math> and <math>E_n(x,\tfrac{1}{4}) = 2^{-n}U_n(x)</math>.

===Other properties===
The curves given by {{math|''y'' {{=}} ''T''<sub>''n''</sub>(''x'')}}, or equivalently, by the parametric equations {{math|''y'' {{=}} ''T''<sub>''n''</sub>(cos ''θ'') {{=}} cos ''nθ''}}, {{math|''x'' {{=}} cos ''θ''}}, are a special case of [[Lissajous curve]]s with frequency ratio equal to {{mvar|n}}.

Similar to the formula:

<math display="block">T_n(\cos\theta) = \cos(n\theta),</math>

we have the analogous formula:

<math display="block">T_{2n+1}(\sin\theta) = (-1)^n \sin\left(\left(2n+1\right)\theta\right).</math>

For {{math|''x'' ≠ 0}}:

<math display="block">T_n\!\left(\frac{x + x^{-1}}{2}\right) = \frac{x^n+x^{-n}}{2}</math>

and:

<math display="block">x^n = T_n\! \left(\frac{x+x^{-1}}{2}\right)
+ \frac{x-x^{-1}}{2}\ U_{n-1}\!\left(\frac{x+x^{-1}}{2}\right),</math>
which follows from the fact that this holds by definition for {{math|1=''x'' = ''e<sup>iθ</sup>''}}.

There are relations between [[Legendre polynomial]]s and Chebyshev polynomials

<math>\sum_{k=0}^{n}P_{k}\left(x\right)T_{n-k}\left(x\right) = \left(n+1\right)P_{n}\left(x\right)</math>

<math>\sum_{k=0}^{n}P_{k}\left(x\right)P_{n-k}\left(x\right) = U_{n}\left(x\right)</math>

These identities can be proven using generating functions and discrete convolution

====Chebyshev polynomials as determinants====

From their definition by recurrence it follows that the Chebyshev polynomials can be obtained as [[determinant]]s of special [[tridiagonal matrix|tridiagonal matrices]] of size <math>k \times k</math>:

<math display="block">T_k(x) = \det
\begin{bmatrix}
         x &      1 &      0 & \cdots & 0      \\
         1 &     2x &      1 & \ddots & \vdots \\
         0 &      1 &     2x & \ddots & 0      \\
    \vdots & \ddots & \ddots & \ddots & 1      \\
         0 & \cdots &      0 &      1 & 2x
\end{bmatrix},</math>
and similarly for <math>U_k</math>.

==Examples==

===First kind===
[[File:Chebyshev Polynomials of the 1st Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the first kind in the domain {{math|−1 < ''x'' < 1}}:
The flat <span style="color:purple;">{{math|''T''<sub>0</sub>}}</span>, <span style="color:red;">{{math|''T''<sub>1</sub>}}</span>, <span style="color:blue;">{{math|''T''<sub>2</sub>}}</span>, <span style="color:green;">{{math|''T''<sub>3</sub>}}</span>, <span style="color:orange;">{{math|''T''<sub>4</sub>}}</span> and <span style="color:black;">{{math|''T''<sub>5</sub>}}</span>.]]

The first few Chebyshev polynomials of the first kind are {{OEIS2C|A028297}}

<math display="block"> \begin{align}
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 \\
T_{10}(x) &= 512x^{10} - 1280x^8 + 1120x^6 - 400x^4 + 50x^2-1
\end{align}</math>

===Second kind===
[[File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the second kind in the domain {{math|−1 < ''x'' < 1}}: The flat <span style="color:purple;">{{math|''U''<sub>0</sub>}}</span>, <span style="color:red;">{{math|''U''<sub>1</sub>}}</span>, <span style="color:blue;">{{math|''U''<sub>2</sub>}}</span>, <span style="color:green;">{{math|''U''<sub>3</sub>}}</span>, <span style="color:orange;">{{math|''U''<sub>4</sub>}}</span> and <span style="color:black;">{{math|''U''<sub>5</sub>}}</span>.
Although not visible in the image, {{math|1=''U''<sub>''n''</sub>(1) = ''n''&nbsp;+&thinsp;1}} and {{math|1=''U''<sub>''n''</sub>(−1) = (''n''&nbsp;+&thinsp;1)(−1)<sup>''n''</sup>}}.]]

The first few Chebyshev polynomials of the second kind are {{OEIS2C|A053117}}

<math display="block">\begin{align}
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 \\
U_{10}(x) &= 1024x^{10} - 2304 x^8 + 1792 x^6 - 560 x^4 + 60 x^2-1
\end{align}</math>

==As a basis set==
[[Image:ChebyshevExpansion.png|thumb|right|240px|The non-smooth function (top) {{math|1=''y'' = −''x''<sup>3</sup>''H''(−''x'')}}, where {{mvar|H}} is the [[Heaviside step function]], and (bottom) the 5th partial sum of its Chebyshev expansion. 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 an [[Hilbert space#Orthonormal bases|orthonormal basis]], so that a function in the same space can, on {{math|−1 ≤ ''x'' ≤ 1}}, be expressed via the expansion:<ref name=boyd>{{cite book|title = Chebyshev and Fourier Spectral Methods|first = John P.|last = Boyd|isbn = 0-486-41183-4|edition = second|year = 2001|publisher = Dover|url = http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf|access-date = 2009-03-19|archive-url = https://web.archive.org/web/20100331183829/http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf|archive-date = 2010-03-31|url-status = dead}}</ref>

<math display="block">f(x) = \sum_{n = 0}^\infty a_n T_n(x).</math>

Furthermore, as mentioned previously, the Chebyshev polynomials form an [[orthogonal]] basis which (among other things) implies that the coefficients {{math|''a''<sub>''n''</sub>}} 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.<ref name=boyd/> These attributes include:

* The Chebyshev polynomials form a [[Complete metric space|complete]] orthogonal system.
* The Chebyshev series converges to {{math|''f''(''x'')}} if the function is [[piecewise]] [[Smooth function|smooth]] and [[Continuous function|continuous]]. The smoothness requirement can be relaxed in most cases{{snd}} as long as there are a finite number of discontinuities in {{math|''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]],<ref name=boyd/> often in favor of trigonometric series due to generally faster convergence for continuous functions ([[Gibbs' phenomenon]] is still a problem).

The [[Chebfun]] software package supports function manipulation based on their expansion in the Chebyshev basis.

===Example 1===
Consider the Chebyshev expansion of {{math|log(1&thinsp;+&nbsp;''x'')}}. One can express:

<math display="block"> \log(1+x) = \sum_{n = 0}^\infty a_n T_n(x)~. </math>

One can find the coefficients {{math|''a<sub>n</sub>''}} either through the application of an inner product or by the discrete orthogonality condition. For the inner product:

<math display="block">\int_{-1}^{+1}\,\frac{T_m(x)\,\log(1 + x)}{\sqrt{1-x^2}}\,\mathrm{d}x = \sum_{n=0}^{\infty}a_n\int_{-1}^{+1}\frac{T_m(x)\,T_n(x)}{\sqrt{1-x^2}}\,\mathrm{d}x,</math>
which gives:
<math display="block">a_n = \begin{cases}
-\log 2            & \text{ for }~ n = 0, \\
\frac{-2(-1)^n}{n} & \text{ for }~ n > 0.
\end{cases}</math>

Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for ''approximate'' coefficients:

<math display="block">a_n \approx \frac{\,2-\delta_{0n}\,}{N}\,\sum_{k=0}^{N-1}T_n(x_k)\,\log(1+x_k),</math>

where {{mvar|δ<sub>ij</sub>}} is the [[Kronecker delta]] function and the {{mvar|x<sub>k</sub>}} are the {{mvar|N}} Gauss–Chebyshev zeros of {{math|''T''<sub>''N''&thinsp;</sub>(''x'')}}:

<math display="block"> x_k = \cos\left(\frac{\pi\left(k+\tfrac{1}{2}\right)}{N}\right) .</math>

For any {{mvar|N}}, these approximate coefficients provide an exact approximation to the function at {{mvar|x<sub>k</sub>}} with a controlled error between those points. The exact coefficients are obtained with {{math|1=''N'' = ∞}}, thus representing the function exactly at all points in {{closed-closed|−1,1}}. The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients {{mvar|a<sub>n</sub>}} very efficiently through the [[discrete cosine transform]]:

<math display="block">a_n \approx \frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}\cos\left(\frac{n\pi\left(\,k+\tfrac{1}{2}\right)}{N}\right)\log(1+x_k).</math>

===Example 2===
To provide another example:

<math display="block">\begin{align}
\left(1-x^2\right)^\alpha &= -\frac{1}{\sqrt{\pi}} \, \frac{\Gamma\left(\tfrac{1}{2} + \alpha\right)}{\Gamma(\alpha+1)} + 2^{1-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2 \alpha \choose \alpha-n}\,T_{2n}(x) \\[1ex]
 &= 2^{-2\alpha}\,\sum_{n=0} \left(-1\right)^n \, {2\alpha+1 \choose \alpha-n}\,U_{2n}(x).
\end{align}</math>

===Partial sums===
The partial sums of:

<math display="block">f(x) = \sum_{n = 0}^\infty a_n T_n(x)</math>

are very useful in the [[approximation theory|approximation]] of various functions and in the solution of [[differential equation]]s (see [[spectral method]]). Two common methods for determining the coefficients {{mvar|a<sub>n</sub>}} are through the use of the [[inner product]] as in [[Galerkin's method]] and through the use of [[collocation method|collocation]] which is related to [[interpolation]].

As an interpolant, the {{mvar|N}} coefficients of the {{math|(''N''&nbsp;−&thinsp;1)}}st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto<ref>{{Cite web |url=http://www.scottsarra.org/chebyApprox/chebyshevApprox.html |title=Chebyshev Interpolation: An Interactive Tour |access-date=2016-06-02 |archive-url=https://web.archive.org/web/20170318214311/http://www.scottsarra.org/chebyApprox/chebyshevApprox.html |archive-date=2017-03-18 |url-status=dead }}</ref> 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:

<math display="block">x_k = -\cos\left(\frac{k \pi}{N - 1}\right); \qquad k = 0, 1, \dots, N - 1.</math>

===Polynomial in Chebyshev form===
An arbitrary polynomial of degree {{mvar|N}} can be written in terms of the Chebyshev polynomials of the first kind.{{sfn|Mason|Handscomb|2002}} Such a polynomial {{math|''p''(''x'')}} is of the form:

<math display="block">p(x) = \sum_{n=0}^N a_n T_n(x).</math>

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

== Families of polynomials related to Chebyshev polynomials ==
Polynomials denoted <math>C_n(x)</math> and <math>S_n(x)</math> closely related to Chebyshev polynomials are sometimes used. They are defined by:{{sfn|Hochstrasser|1972|p=778}}

<math display="block">C_n(x) = 2T_n\left(\frac{x}{2}\right),\qquad S_n(x) = U_n\left(\frac{x}{2}\right)</math>

and satisfy:

<math display="block">C_n(x) = S_n(x) - S_{n-2}(x).</math>

A. F. Horadam called the polynomials <math>C_n(x)</math> '''Vieta–Lucas polynomials''' and denoted them <math>v_n(x)</math>. He called the polynomials 
<math>S_n(x)</math> '''Vieta–Fibonacci polynomials''' and denoted them {{nowrap|<math>V_n(x)</math>.}}<ref>{{citation|last=Horadam|first=A. F.|title=Vieta polynomials|journal=Fibonacci Quarterly|volume=40|issue=3|year=2002|url=https://www.fq.math.ca/Scanned/40-3/horadam2.pdf|pages=223–232}}</ref> Lists of both sets of polynomials are given in [[François Viète|Viète's]] ''Opera Mathematica'', Chapter IX, Theorems VI and VII.<ref>{{cite book|last=Viète|first=François|title=Francisci Vietae Opera mathematica : in unum volumen congesta ac recognita / opera atque studio Francisci a Schooten|year=1646|publisher=Bibliothèque nationale de France|url=https://gallica.bnf.fr/ark:/12148/bpt6k107597d.pdf}}</ref> The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of <math>i</math> and a shift of index in the case of the latter, equal to [[Fibonacci polynomials|Lucas and Fibonacci polynomials]] {{math|''L''<sub>''n''</sub>}} and {{math|''F''<sub>''n''</sub>}} of imaginary argument.

'''Shifted Chebyshev polynomials''' of the first and second kinds are related to the Chebyshev polynomials by:{{sfn|Hochstrasser|1972|p=778}}

<math display="block">T_n^*(x) = T_n(2x-1),\qquad U_n^*(x) = U_n(2x-1).</math>

When the argument of the Chebyshev polynomial satisfies {{math|2''x'' − 1 ∈ {{closed-closed|−1, 1}}}} the argument of the shifted Chebyshev polynomial satisfies {{math|''x'' ∈ {{closed-closed|0, 1}}}}. Similarly, one can define shifted polynomials for generic intervals {{closed-closed|''a'', ''b''}}.

Around 1990 the terms "third-kind" and "fourth-kind" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name '''airfoil polynomials'''. According to J. C. Mason and G. H. Elliott, the terminology "third-kind" and "fourth-kind" is due to [[Walter Gautschi]], "in consultation with colleagues in the field of orthogonal polynomials."<ref name=MasonElliott1993>{{citation|last1=Mason|first1=J. C.|last2=Elliott | first2=G. H.|title= Near-minimax complex approximation by four kinds of Chebyshev polynomial expansion | journal = J. Comput. Appl. Math. | volume = 46| pages = 291–300| year = 1993| issue = 1–2 | doi = 10.1016/0377-0427(93)90303-S | doi-access = free}}</ref> The '''Chebyshev polynomials of the third kind''' are defined as:

<math display="block">V_n(x)=\frac{\cos\left(\left(n+\frac{1}{2}\right)\theta\right)}{\cos\left(\frac{\theta}{2}\right)}=\sqrt\frac{2}{1+x}T_{2n+1}\left(\sqrt\frac{x+1}{2}\right)</math>
and the '''Chebyshev polynomials of the fourth kind''' are defined as:
<math display="block">W_n(x)=\frac{\sin\left(\left(n+\frac{1}{2}\right)\theta\right)}{\sin\left(\frac{\theta}{2}\right)}=U_{2n}\left(\sqrt\frac{x+1}{2}\right),</math>

where <math>\theta=\arccos x</math>.<ref name=MasonElliott1993/><ref name=DesmaraisBland1995>{{citation|last1=Desmarais|first1=Robert N.|last2=Bland|first2=Samuel R.|title=Tables of properties of airfoil polynomials|publisher=National Aeronautics and Space Administration|work=NASA Reference Publication 1343|url=https://ntrs.nasa.gov/citations/19960001864|year=1995}}</ref>
They coincide with the [[Dirichlet kernel]].

In the airfoil literature <math>V_n(x)</math> and <math>W_n(x)</math> are denoted <math>t_n(x)</math> and <math>u_n(x)</math>. The polynomial families <math>T_n(x)</math>, <math>U_n(x)</math>, <math>V_n(x)</math>, and <math>W_n(x)</math> are orthogonal with respect to the weights:

<math display="block">\left(1-x^2\right)^{-1/2},\quad\left(1-x^2\right)^{1/2},\quad(1-x)^{-1/2}(1+x)^{1/2},\quad(1+x)^{-1/2}(1-x)^{1/2}</math>

and are proportional to Jacobi polynomials <math>P_n^{(\alpha,\beta)}(x)</math> with:<ref name="DesmaraisBland1995" />

<math display="block">(\alpha,\beta)=\left(-\frac{1}{2},-\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(-\frac{1}{2},\frac{1}{2}\right),\quad(\alpha,\beta)=\left(\frac{1}{2},-\frac{1}{2}\right).</math>

All four families satisfy the recurrence <math>p_n(x)=2xp_{n-1}(x)-p_{n-2}(x)</math> with <math>p_0(x) = 1</math>, where <math>p_n = T_n</math>, <math>U_n</math>, <math>V_n</math>, or <math>W_n</math>, but they differ according to whether <math>p_1(x)</math> equals <math>x</math>, <math>2x</math>, <math>2x-1</math>, or {{nowrap|<math>2x+1</math>.}}<ref name=MasonElliott1993/>

=== Even order modified Chebyshev polynomials ===
Some applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications.  Even order [[Chebyshev filter]] designs using equally terminated passive networks are an example of this.<ref name=":022">{{Cite book |last=Saal |first=Rudolf |url=https://archive.org/details/handbuchzumfilte0000saal |title=Handbook of Filter Design |publisher=Allgemeine Elektricitais-Gesellschaft |date=January 1979 |isbn=3-87087-070-2 |edition=1st |location=Munich, Germany |pages=25, 26, 56–61, 116, 117 |language=English, German}}</ref>  However, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect.  Such modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials.  Even order modified Chebyshev polynomials may be created from the [[Chebyshev nodes]] in the same manner as standard Chebyshev polynomials.

<math display="block">P_N = \prod_{i=1}^N(x-C_i)</math>

where

* <math>P_N</math> is an ''N''-th order Chebyshev polynomial
* <math>C_i</math> is the ''i''-th Chebyshev node

In the case of even order modified Chebyshev polynomials, the [[Chebyshev nodes#Even order modified Chebyshev nodes|even order modified Chebyshev nodes]] are used to construct the even order modified  Chebyshev polynomials.

<math display="block">Pe_N = \prod_{i=1}^N(x-Ce_i)</math>

where

* <math>P e_N</math> is an ''N''-th order even order modified Chebyshev polynomial
* <math>Ce_i</math> is the ''i''-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from the [[Chebyshev polynomials#Examples|example above]] is <math>X^4-X^2+.125
</math>, which by inspection contains no roots of zero.  Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of <math>X^4-.828427X^2
</math>, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.

==See also==
{{Portal|Mathematics}}
*[[Chebyshev rational functions]]
*[[Function approximation]]
*[[Discrete Chebyshev transform]]
*[[Markov brothers' inequality]]

==References==
{{Reflist|30em}}

==Sources==
* {{cite book
 | last = Hochstrasser | first = Urs W.
 | chapter = Orthogonal Polynomials
 | year = 1972 |orig-year = 1964
 | edition = 10th printing, with corrections; first
 | editor1-first=Milton |editor1-last=Abramowitz |editor1-link=Milton Abramowitz
 | editor2-first=Irene |editor2-last=Stegun |editor2-link=Irene Stegun
 | title = Handbook of Mathematical Functions
 | title-link = Abramowitz and Stegun
 | location=Washington D.C.
 | publisher=National Bureau of Standards
 | lccn=64-60036
 | mr=0167642
 | at = Ch. 22, {{pgs|771–792}}
 | chapter-url = https://personal.math.ubc.ca/~cbm/aands/page_771.htm
}} Reprint: 1983. New York: Dover. {{isbn|978-0-486-61272-0}}.
* {{cite book |last1=Bateman |first1=Harry |author1-link=Harry Bateman |author2=Bateman Manuscript Project |editor1-last=Erdélyi |editor1-first=Arthur |editor1-link=Arthur Erdélyi |title=Higher Transcendental Functions |chapter=Tchebichef polynomials |others=Research associates: [[Hans Heinrich Wilhelm Magnus|W. Magnus]], {{ill|F. Oberhettinger|de|Fritz Oberhettinger}}, [[Francesco Giacomo Tricomi|F. Tricomi]] <!-- others involved include David Bertin, Watson B. Fulks, Albert Raymond Harvey, Donald L. Thomsen Jr., Maria A. Weber; Eoin Laird Whitney, Rosemarie Stampfel --> |year=1953 |place=New York |publisher=McGraw-Hill |edition=1st |volume=2 |lccn=53-5555 |id=Caltech eprint [https://authors.library.caltech.edu/records/cnd32-h9x80 43491] |chapter-url=https://archive.org/details/highertranscende02bate |chapter-url-access=limited |at=§ 10.11, {{pgs|183–187}} }} Reprint: 1981. Melbourne, FL: Krieger. {{ISBN|0-89874-069-X}}.
*{{cite book
 |last1=Mason |first1=J. C.
 |last2=Handscomb |first2=D.C.
 |year=2002
 |url={{Google books|8FHf0P3to0UC|Chebyshev Polynomials|plainurl=yes}}
 |title=Chebyshev Polynomials
 |publisher=Chapman and Hall/CRC
 |isbn=978-1-4200-3611-4
 |doi=10.1201/9781420036114
}}

== Further reading ==

{{refbegin|35em}}
* {{cite journal
 |last1=Dette |first1=Holger
 |year=1995
 |title= A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials
 |journal=Proceedings of the Edinburgh Mathematical Society
 |volume=38 |number=2 |pages=343–355
 |arxiv=math/9406222
 |doi=10.1017/S001309150001912X
 }}
* {{cite journal
 |first1=David |last1=Elliott
 |year=1964
 |title=The evaluation and estimation of the coefficients in the Chebyshev Series expansion of a function
 |journal=Math. Comp.
 | volume=18 | number=86 | pages=274–284
 |mr=0166903
 |doi=10.1090/S0025-5718-1964-0166903-7 |doi-access=free
}}
*{{cite journal
 | last1=Eremenko | first1=A.
 |last2=Lempert  |first2=L.
 |year =1994
 | url=http://www.math.purdue.edu/~eremenko/dvi/lempert.pdf
 |doi=10.1090/S0002-9939-1994-1207536-1
 |mr=1207536
 |title=An Extremal Problem For Polynomials
 | journal=[[Proceedings of the American Mathematical Society]]
 |volume=122 |number=1 |pages=191–193
 |doi-access=free
 }}
*{{cite journal
 | first1=M. A. | last1=Hernandez
 | title=Chebyshev's approximation algorithms and applications
 | year=2001
 | journal=Computers & Mathematics with Applications
 | volume=41 | issue=3–4 |pages=433–445
 | doi=10.1016/s0898-1221(00)00286-8
 | doi-access=free
 }}
*{{cite book
 |first1=J. C. |last1=Mason
 |title=Rational Approximation and Interpolation
 |year=1984
 |chapter=Some properties and applications of Chebyshev polynomial and rational approximation
 |series=Lecture Notes in Mathematics
 |volume=1105 |pages=27–48
 |doi=10.1007/BFb0072398 |doi-access=free |isbn=978-3-540-13899-0
}}
*{{dlmf
 |first=Tom H. |last=Koornwinder
 |first2=Roderick S. C. |last2= Wong
 |first3=Roelof |last3=Koekoek
 |first4=René F. |last4=Swarttouw
 |title=Orthogonal Polynomials
 |id=18
}}
* {{cite web
 |last=Remes |first=Eugene
 |title= On an Extremal Property of Chebyshev Polynomials
 |url=https://www.math.technion.ac.il/hat/fpapers/remeztrans.pdf
}}
*{{cite journal
 |first1=Herbert E. |last1=Salzer
 | year=1976
 | title=Converting interpolation series into Chebyshev series by recurrence formulas
 |journal=Mathematics of Computation
 | volume=30|number=134 | pages=295–302
 |doi=10.1090/S0025-5718-1976-0395159-3 |doi-access=free |mr=0395159
}}
*{{cite journal
 | first1=R.E. | last1=Scraton
 | year=1969
 | title=The Solution of integral equations in Chebyshev series
 | journal=Mathematics of Computation
 | volume=23 | number=108 |pages=837–844
 | mr=0260224 | doi=10.1090/S0025-5718-1969-0260224-4 |doi-access=free
}}
*{{cite journal
 |first1=Lyle B. |last1=Smith
 |year=1966
 |title=Computation of Chebyshev series coefficients
 |id=Algorithm 277
 |journal=Comm. ACM
 |volume=9 | number=2 | pages=86–87
 |doi=10.1145/365170.365195|s2cid=8876563
 |doi-access=free
 }}
*{{eom|title=Chebyshev polynomials|first=P. K.|last= Suetin}}
{{refend}}

==External links==
*{{Commons category-inline}}
* {{MathWorld
 |title=Chebyshev polynomial[s] of the first kind
 |urlname=ChebyshevPolynomialoftheFirstKind
}}
* {{cite web
 |first=John H. |last=Mathews
 |year=2003
 |title=Module for Chebyshev polynomials
 |series=Course notes for Math 340 ''Numerical Analysis'' & Math 440 ''Advanced Numerical Analysis''
 |publisher=California State University
 |place=Fullerton, CA
 |department=Department of Mathematics
 |url=http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html
 |url-status=dead |access-date=2020-08-17 |df=dmy-all
 |archive-url=https://web.archive.org/web/20070529221407/http://math.fullerton.edu/mathews/n2003/ChebyshevPolyMod.html
 |archive-date=2007-05-29
}}
* {{cite web
 |title=Numerical computing with functions
 |url=http://www.chebfun.org
 |website=The Chebfun Project 
}}
* {{cite web
 |title=Is there an intuitive explanation for an extremal property of Chebyshev polynomials?
 |website=Math Overflow
 |id=Question&nbsp;25534
 |url=https://mathoverflow.net/q/25534
}}
* {{cite web
 |title=Chebyshev polynomial evaluation and the Chebyshev transform
 |website=Boost
 |series=Math
 |url=https://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/sf_poly/chebyshev.html 
}}


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