Chebyshev polynomials

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

$$\backslash begin\{align\}$$

T_0(x) & = 1, \\

T_1(x) & = x, \\

T_{n+1}(x) & = 2 x\,T_n(x) - T_{n-1}(x).

\end{align}

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

$$\backslash begin\{align\}$$

U_0(x) & = 1, \\

U_1(x) & = 2 x, \\

U_{n+1}(x) & = 2 x\,U_n(x) - U_{n-1}(x),

\end{align}

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

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

$$T\_n(\backslash cos\backslash theta)\; =\; \backslash cos(n\backslash theta)$$

and

$$U\_n(\backslash cos\backslash theta)\; =\; \backslash frac\{\backslash sin\backslash big((n\; +\; 1)\backslash theta\backslash big)\}\{\backslash sin\backslash theta\},$$

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,

$$z^n\; =\; T\_n(a)\; +\; ib\; U\_\{n-1\}(a).$$

Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.{{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 }}

That $\backslash cos(nx)$ is an $m$th-degree polynomial in $\backslash cos(x)$ can be seen by observing that $\backslash cos(nx)$ is the real part of one side of de Moivre's formula:

$$\backslash cos\; n\; \backslash theta\; +\; i\; \backslash sin\; n\; \backslash theta\; =\; (\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta)^n.$$

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

For $x$ outside the interval [-1,1], the above definition implies

$$T\_n(x)\; =\; \backslash begin\{cases\}$$

\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}

Chebyshev polynomials can also be characterized by the following theorem:{{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}}

If $F\_n(x)$ is a family of monic polynomials with coefficients in a field of characteristic $0$ such that $\backslash deg\; F\_n(x)\; =\; n$ and $F\_m(F\_n(x))\; =\; F\_n(F\_m(x))$ for all

$m$ and $n$, then, up to a simple change of variables, either $F\_n(x)\; =\; x^n$ for all $n$ or

$F\_n(x)\; =\; 2\backslash cdot\; T\_n(x/2)$ for all $n$.

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

$$T\_n(x)^2\; -\; \backslash left(x^2\; -\; 1\backslash right)\; U\_\{n-1\}(x)^2\; =\; 1$$

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

$$T\_n(x)\; +\; U\_\{n-1\}(x)\backslash ,\backslash sqrt\{x^2-1\}\; =\; \backslash left(x\; +\; \backslash sqrt\{x^2-1\}\backslash right)^n~.$$

The ordinary generating function for $T\_n$ is

$$\backslash sum\_\{n=0\}^\backslash infty\; T\_n(x)\backslash ,t^n\; =\; \backslash frac\{1\; -\; tx\}\{1\; -\; 2tx\; +\; t^2\}.$$

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

$$\backslash begin\{align\}$$

\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}

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

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

The ordinary generating function for {{mvar|U_n}} is

$$\backslash sum\_\{n=0\}^\backslash infty\; U\_n(x)\backslash ,t^n\; =\; \backslash frac\{1\}\{1\; -\; 2tx\; +\; t^2\},$$

and the exponential generating function is

\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).

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences $\backslash tilde\; V\_n(P,Q)$ and $\backslash tilde\; U\_n(P,Q)$ with parameters $P=2x$ and $Q=1$:

$$\backslash begin\{align\}$$

{\tilde U}_n(2x,1) &= U_{n-1}(x), \\

{\tilde V}_n(2x,1) &= 2\, T_n(x).

\end{align}

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]}}

$$\backslash begin\{align\}$$

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}

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

$$T\_n(x)\; =\; \backslash frac\{1\}\{2\}\; \backslash big(U\_n(x)\; -\; U\_\{n-2\}(x)\backslash big).$$

Using this formula iteratively gives the sum formula:

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}

while replacing $U\_n(x)$ and $U\_\{n-2\}(x)$ using the derivative formula for $T\_n(x)$ gives the recurrence relationship for the derivative of $T\_n$:

$$2\backslash ,T\_n(x)\; =\; \backslash frac\{1\}\{n+1\}\backslash ,\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\; T\_\{n+1\}(x)\; -\; \backslash frac\{1\}\{n-1\}\backslash ,\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash ,\; T\_\{n-1\}(x),\; \backslash qquad\; n=2,3,\backslash ldots$$

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

Turán's inequalities for the Chebyshev polynomials are:{{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}}

$$\backslash begin\{align\}$$

T_n(x)^2 - T_{n-1}(x)\,T_{n+1}(x)&= 1-x^2 > 0 &&\text{ for } -1

U_n(x)^2 - U_{n-1}(x)\,U_{n+1}(x)&= 1 > 0~.

\end{align}

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}}

$$\backslash begin\{align\}$$

\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}

where integrals are considered as principal value.

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}}

$$\backslash begin\{align\}$$

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}

The two are equivalent because $\backslash textstyle\; \backslash bigl(x\; +\; \backslash sqrt\{x^2\; -\; 1\}\backslash !~\backslash bigr)\backslash bigl(x\; -\; \backslash sqrt\{x^2\; -\; 1\}\backslash !~\backslash bigr)\; =\; 1$.

An explicit form of the Chebyshev polynomial in terms of monomials $x^k$ follows from de Moivre's formula:

$$T\_n(\backslash cos(\backslash theta))\; =\; \backslash operatorname\{Re\}(\backslash cos\; n\; \backslash theta\; +\; i\; \backslash sin\; n\; \backslash theta)\; =\; \backslash operatorname\{Re\}((\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta)^n),$$

where $\backslash mathrm\{Re\}$ denotes the real part of a complex number. Expanding the formula, one gets

$$(\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta)^n\; =\; \backslash sum\backslash limits\_\{j=0\}^n\; \backslash binom\{n\}\{j\}\; i^j\; \backslash sin^j\; \backslash theta\; \backslash cos^\{n-j\}\; \backslash theta.$$

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

$$\backslash cos\; n\; \backslash theta\; =\; \backslash sum\backslash limits\_\{j=0\}^\{\backslash lfloor\; n\; /\; 2\; \backslash rfloor\}\; \backslash binom\{n\}\{2j\}\; (\backslash cos^2\; \backslash theta\; -\; 1)^j\; \backslash cos^\{n-2j\}\; \backslash theta,$$

which in turn means that

$$T\_n(x)\; =\; \backslash sum\backslash limits\_\{j=0\}^\{\backslash lfloor\; n\; /\; 2\; \backslash rfloor\}\; \backslash binom\{n\}\{2j\}\; (x^2-1)^j\; x^\{n-2j\}.$$

This can be written as a {{math|₂F₁}} hypergeometric function:

$$\backslash begin\{align\}$$

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}

with inverse{{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}}{{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

}}

$$x^n\; =\; 2^\{1-n\}\backslash mathop\{\{\backslash sum\}\text{'}\}^n\_\{j=0\backslash atop\; j\; \backslash equiv\; n\; \backslash pmod\; 2\}\; \backslash !\backslash !\backslash binom\{n\}\{\backslash tfrac\{n-j\}\{2\}\}\backslash !\backslash ;T\_j(x),$$

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

A related expression for $T\_n$ as a sum of monomials with binomial coefficients and powers of two is

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}.

Similarly, $U\_n$ can be expressed in terms of hypergeometric functions:

$$\backslash begin\{align\}$$

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}

$$\backslash begin\{align\}$$

T_n(-x) &= (-1)^n\, T_n(x),\\[1ex]

U_n(-x) &= (-1)^n\, U_n(x).

\end{align}

That is, Chebyshev polynomials of even order have even symmetry and therefore contain only even powers of $x$. Chebyshev polynomials of odd order have odd symmetry and therefore contain only odd powers of $x$.

A Chebyshev polynomial of either kind with degree {{mvar|n}} has {{mvar|n}} different simple roots, 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:

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

one can show that the roots of $T\_n$ are:

$$x\_k\; =\; \backslash cos\backslash left(\backslash frac\{\backslash pi(k+1/2)\}\{n\}\backslash right),\backslash quad\; k=0,\backslash ldots,n-1.$$

Similarly, the roots of $U\_n$ are:

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

The extrema of $T\_n$ on the interval $-1\backslash leq\; x\backslash leq\; 1$ are located at:

$$x\_k\; =\; \backslash cos\backslash left(\backslash frac\{k\}\{n\}\backslash pi\backslash right),\backslash quad\; k=0,\backslash ldots,n.$$

One unique property of the Chebyshev polynomials of the first kind is that on the interval $-1\backslash leq\; x\backslash leq\; 1$ 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:

$$\backslash begin\{align\}$$

T_n(1) &= 1 \\

T_n(-1) &= (-1)^n \\

U_n(1) &= n+1 \\

U_n(-1) &= (-1)^n (n+1).

\end{align}

The extrema of $T\_n(x)$ on the interval $-1\; \backslash leq\; x\; \backslash leq\; 1$ where $n>0$ are located at $n+1$ values of $x$. They are $\backslash pm\; 1$, or $\backslash cos\backslash left(\backslash frac\{2\backslash pi\; k\}\{d\}\backslash right)$ where $d\; >\; 2$, $d\; \backslash ;|\backslash ;\; 2n$, and $(k,\; d)\; =\; 1$, i.e., $k$ and $d$ are relatively prime numbers.

Specifically (Minimal polynomial of 2cos(2pi/n){{cite journal |first1=Y. Z. |last1=Gürtaş |title=Chebyshev Polynomials and the minimal polynomial of $\backslash cos\; (2\; \backslash pi/n)$ |year=2017 |journal=American Mathematical Monthly |volume=124 |number=1 |pages=74–78 |doi=10.4169/amer.math.monthly.124.1.74|s2cid=125797961 }}{{cite journal |first1=D. A. |last1=Wolfram |title=Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of $\backslash cos\; (2\; \backslash pi\; /d\; )$ |year=2022 |journal=American Mathematical Monthly |volume=129 |number=2 |pages=172–176 |doi=10.1080/00029890.2022.2005391|s2cid=245808448 }}) when $n$ is even:

• $T\_n(x)\; =\; 1$ if $x\; =\; \backslash pm\; 1$, or $d\; >\; 2$ and $2n/d$ is even. There are $n/2\; +\; 1$ such values of $x$.
• $T\_n(x)\; =\; -1$ if $d\; >\; 2$ and $2n/d$ is odd. There are $n/2$ such values of $x$.

When $n$ is odd:

• $T\_n(x)\; =\; 1$ if $x\; =\; 1$, or $d\; >\; 2$ and $2n/d$ is even. There are $(n+1)/2$ such values of $x$.
• $T\_n(x)\; =\; -1$ if $x\; =\; -1$, or $d\; >\; 2$ and $2n/d$ is odd. There are $(n+1)/2$ such values of $x$.

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

$$\backslash begin\{align\}$$

\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}

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

$$\backslash begin\{align\}$$

\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}

More generally,

$$\backslash left.\backslash frac\{\backslash mathrm\{d\}^p\; T\_n\}\{\backslash mathrm\{d\}x^p\}\; \backslash right|\_\{x\; =\; \backslash pm\; 1\}\; \backslash !\backslash !\; =\; (\backslash pm\; 1)^\{n+p\}\backslash prod\_\{k=0\}^\{p-1\}\backslash frac\{n^2-k^2\}\{2k+1\}~,$$

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

Also, we have:

$$\backslash frac\{\backslash mathrm\{d\}^p\}\{\backslash mathrm\{d\}x^p\}\backslash ,T\_n(x)\; =\; 2^p\backslash ,n\backslash mathop\{\{\backslash sum\}\text{'}\}\_\{0\backslash leq\; k\backslash leq\; n-p\backslash atop\; k\; \backslash ,\backslash equiv\backslash ,\; n-p\; \backslash pmod\; 2\}$$

\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,

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_n}} implies that:

$$\backslash int\; U\_n\backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{T\_\{n\; +\; 1\}\}\{n\; +\; 1\}$$

and the recurrence relation for the first kind polynomials involving derivatives establishes that for $n\backslash geq\; 2$:

$$\backslash int\; T\_n\backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{1\}\{2\}\backslash ,\backslash left(\backslash frac\{T\_\{n\; +\; 1\}\}\{n\; +\; 1\}\; -\; \backslash frac\{T\_\{n\; -\; 1\}\}\{n\; -\; 1\}\backslash right)\; =\; \backslash frac\{n\backslash ,T\_\{n\; +\; 1\}\}\{n^2\; -\; 1\}\; -\; \backslash frac\{x\backslash ,T\_n\}\{n\; -\; 1\}.$$

The last formula can be further manipulated to express the integral of $T\_n$ as a function of Chebyshev polynomials of the first kind only:

$$\backslash begin\{align\}$$

\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}

Furthermore, we have:

$$\backslash int\_\{-1\}^1\; T\_n(x)\backslash ,\; \backslash mathrm\{d\}x\; =$$

\begin{cases}

\frac{(-1)^n + 1}{1 - n^2} & \text{ if }~ n \ne 1 \\

0 & \text{ if }~ n = 1.

\end{cases}

The Chebyshev polynomials of the first kind satisfy the relation:

$$T\_m(x)\backslash ,T\_n(x)\; =\; \backslash tfrac\{1\}\{2\}\backslash !\backslash left(T\_\{m+n\}(x)\; +\; T\_$$

which is easily proved from the product-to-sum formula for the cosine:

$$2\; \backslash cos\; \backslash alpha\; \backslash ,\; \backslash cos\; \backslash beta\; =\; \backslash cos\; (\backslash alpha\; +\; \backslash beta)\; +\; \backslash cos\; (\backslash alpha\; -\; \backslash beta).$$

For $n=1$ this results in the already known recurrence formula, just arranged differently, and with $n=2$ it forms the recurrence relation for all even or all odd 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:

$$\backslash begin\{align\}$$

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}

The polynomials of the second kind satisfy the similar relation:

$$T\_m(x)\backslash ,U\_n(x)\; =\; \backslash begin\{cases\}$$

\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}

(with the definition $U\_\{-1\}\backslash equiv\; 0$ by convention ). They also satisfy:

$$U\_m(x)\backslash ,U\_n(x)\; =\; \backslash sum\_\{k=0\}^n\backslash ,U\_\{m-n+2k\}(x)\; =\; \backslash sum\_\backslash underset\{\backslash text\{\; step\; 2\; \}\}\{p=m-n\}^\{m+n\}\; U\_p(x)~.$$

for $m\backslash geq\; n$.

For $n=2$ this recurrence reduces to:

$$U\_\{m+2\}(x)\; =\; U\_2(x)\backslash ,U\_m(x)\; -\; U\_m(x)\; -\; U\_\{m-2\}(x)\; =\; U\_m(x)\backslash ,\backslash big(U\_2(x)\; -\; 1\backslash big)\; -\; U\_\{m-2\}(x)~,$$

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

The trigonometric definitions of $T\_n$ and $U\_n$ imply the composition or nesting properties:{{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}}

$$\backslash begin\{align\}$$

T_{mn}(x) &= T_m(T_n(x)),\\

U_{mn-1}(x) &= U_{m-1}(T_n(x))U_{n-1}(x).

\end{align}

For $T\_\{mn\}$ the order of composition may be reversed, making the family of polynomial functions $T\_n$ a commutative semigroup under composition.

Since $T\_m(x)$ is divisible by $x$ if $m$ is odd, it follows that $T\_\{mn\}(x)$ is divisible by $T\_n(x)$ if $m$ is odd. Furthermore, $U\_\{mn-1\}(x)$ is divisible by >math>U_{n-1}(x), and in the case that $m$ is even, divisible by $T\_n(x)U\_\{n-1\}(x)$.

Both $T\_n$ and $U\_n$ form a sequence of orthogonal polynomials. The polynomials of the first kind $T\_n$ are orthogonal with respect to the weight:

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

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

$$\backslash int\_\{-1\}^1\; T\_n(x)\backslash ,T\_m(x)\backslash ,\backslash frac\{\backslash mathrm\{d\}x\}\{\backslash sqrt\{1-x^2\}\}\; =$$

\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}

This can be proven by letting $x=\backslash cos(\backslash theta)$ and using the defining identity $T\_n(\backslash cos(\backslash theta)=\backslash cos(n\backslash theta)$.

Similarly, the polynomials of the second kind {{mvar|U_n}} are orthogonal with respect to the weight:

$$\backslash sqrt\{1-x^2\}$$

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

$$\backslash int\_\{-1\}^1\; U\_n(x)\backslash ,U\_m(x)\backslash ,\backslash sqrt\{1-x^2\}\; \backslash ,\backslash mathrm\{d\}x\; =$$

\begin{cases}

0 & ~\text{ if }~ n \ne m, \\[5mu]

\frac{\pi}{2} & ~\text{ if }~ n = m.

\end{cases}

(The measure $\backslash sqrt\{1-x^2\}\backslash ,\; dx$ is, to within a normalizing constant, the Wigner semicircle distribution.)

These orthogonality properties follow from the fact that the Chebyshev polynomials solve the Chebyshev differential equations:

$$\backslash begin\{align\}$$

(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}

which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. (Another way to define the Chebyshev polynomials is as the solutions to those equations.)

The $T\_n$ also satisfy a discrete orthogonality condition:

$$\backslash sum\_\{k=0\}^\{N-1\}\{T\_i(x\_k)\backslash ,T\_j(x\_k)\}\; =$$

\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}

where $N$ is any integer greater than $\backslash max(i,j)$,{{sfn|Mason|Handscomb|2002}} and the $x\_k$ are the $N$ Chebyshev nodes (see above) of $T\_N(x)$:

$$x\_k\; =\; \backslash cos\backslash left(\backslash pi\backslash ,\backslash frac\{2k+1\}\{2N\}\backslash right)\; \backslash quad\; ~\backslash text\{\; for\; \}~\; k\; =\; 0,\; 1,\; \backslash dots,\; N-1.$$

For the polynomials of the second kind and any integer $N>i+j$ with the same Chebyshev nodes $x\_k$, there are similar sums:

$$\backslash sum\_\{k=0\}^\{N-1\}\{U\_i(x\_k)\backslash ,U\_j(x\_k)\backslash left(1-x\_k^2\backslash right)\}\; =$$

\begin{cases}

0 & \text{ if }~ i \ne j, \\[5mu]

\frac{N}{2} & \text{ if }~ i = j,

\end{cases}

and without the weight function:

$$\backslash sum\_\{k=0\}^\{N-1\}\{\; U\_i(x\_k)\; \backslash ,\; U\_j(x\_k)\; \}\; =$$

\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}

For any integer $N>i+j$, based on the $N$} zeros of $U\_N(x)$:

$$y\_k\; =\; \backslash cos\backslash left(\backslash pi\backslash ,\backslash frac\{k+1\}\{N+1\}\backslash right)\; \backslash quad\; ~\backslash text\{\; for\; \}~\; k=0,\; 1,\; \backslash dots,\; N-1,$$

one can get the sum:

$$\backslash sum\_\{k=0\}^\{N-1\}\{U\_i(y\_k)\backslash ,U\_j(y\_k)(1-y\_k^2)\}\; =$$

\begin{cases}

0 & ~\text{ if } i \ne j, \\[5mu]

\frac{N+1}{2} & ~\text{ if } i = j,

\end{cases}

and again without the weight function:

$$\backslash sum\_\{k=0\}^\{N-1\}\{U\_i(y\_k)\backslash ,U\_j(y\_k)\}\; =$$

\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}

For any given $n\backslash geq\; 1$, among the polynomials of degree $n$ with leading coefficient 1 (monic polynomials):

$$f(x)\; =\; \backslash frac\{1\}\{\backslash ,2^\{n-1\}\backslash ,\}\backslash ,T\_n(x)$$

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

This maximal absolute value is:

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

and $|f(x)|$ reaches this maximum exactly $n+1$ times at:

$$x\; =\; \backslash cos\; \backslash frac\{k\backslash pi\}\{n\}\backslash quad\backslash text\{for\; \}0\; \backslash le\; k\; \backslash le\; n.$$

{{Math proof | proof =

Let's assume that $w\_n(x)$ is a polynomial of degree $n$ with leading coefficient 1 with maximal absolute value on the interval {{closed-closed|−1, 1}} less than {{math|1 / 2^n − 1}}.

Define

$$f\_n(x)\; =\; \backslash frac\{1\}\{\backslash ,2^\{n-1\}\backslash ,\}\backslash ,T\_n(x)\; -\; w\_n(x)$$

Because at extreme points of {{mvar|T_n}} we have

$$\backslash begin\{align\}$$

|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}

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

}}

By the equioscillation theorem, among all the polynomials of degree {{math|≤ n}}, the polynomial {{mvar|f}} minimizes {{math|{{norm| f }}_∞}} on {{closed-closed|−1, 1}} if and only if there are {{math|n + 2}} points {{math|−1 ≤ x₀ < x₁ < ⋯ < x_{n + 1} ≤ 1}} such that {{math|1={{abs| f(x_i)}} = {{norm| f }}_∞}}.

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| f }}}} 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).

The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials $C\_n^\{(\backslash lambda)\}(x)$, which themselves are a special case of the Jacobi polynomials $P\_n^\{(\backslash alpha,\backslash beta)\}(x)$:

$$\backslash begin\{align\}$$

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}

Chebyshev polynomials are also a special case of Dickson polynomials:

$$D\_n(2x\backslash alpha,\backslash alpha^2)=\; 2\backslash alpha^\{n\}T\_n(x)\; \backslash ,$$

$$E\_n(2x\backslash alpha,\backslash alpha^2)=\; \backslash alpha^\{n\}U\_n(x).\; \backslash ,$$

In particular, when $\backslash alpha=\backslash tfrac\{1\}\{2\}$, they are related by $D\_n(x,\backslash tfrac\{1\}\{4\})\; =\; 2^\{1-n\}T\_n(x)$ and $E\_n(x,\backslash tfrac\{1\}\{4\})\; =\; 2^\{-n\}U\_n(x)$.

The curves given by {{math|y {{=}} T_n(x)}}, or equivalently, by the parametric equations {{math|y {{=}} T_n(cos θ) {{=}} cos nθ}}, {{math|x {{=}} cos θ}}, are a special case of Lissajous curves with frequency ratio equal to {{mvar|n}}.

Similar to the formula:

$$T\_n(\backslash cos\backslash theta)\; =\; \backslash cos(n\backslash theta),$$

we have the analogous formula:

$$T\_\{2n+1\}(\backslash sin\backslash theta)\; =\; (-1)^n\; \backslash sin\backslash left(\backslash left(2n+1\backslash right)\backslash theta\backslash right).$$

For {{math|x ≠ 0}}:

$$T\_n\backslash !\backslash left(\backslash frac\{x\; +\; x^\{-1\}\}\{2\}\backslash right)\; =\; \backslash frac\{x^n+x^\{-n\}\}\{2\}$$

and:

$$x^n\; =\; T\_n\backslash !\; \backslash left(\backslash frac\{x+x^\{-1\}\}\{2\}\backslash right)$$

+ \frac{x-x^{-1}}{2}\ U_{n-1}\!\left(\frac{x+x^{-1}}{2}\right),

which follows from the fact that this holds by definition for {{math|1=x = e^iθ}}.

There are relations between Legendre polynomials and Chebyshev polynomials

$\backslash sum\_\{k=0\}^\{n\}P\_\{k\}\backslash left(x\backslash right)T\_\{n-k\}\backslash left(x\backslash right)\; =\; \backslash left(n+1\backslash right)P\_\{n\}\backslash left(x\backslash right)$

$\backslash sum\_\{k=0\}^\{n\}P\_\{k\}\backslash left(x\backslash right)P\_\{n-k\}\backslash left(x\backslash right)\; =\; U\_\{n\}\backslash left(x\backslash right)$

These identities can be proven using generating functions and discrete convolution

From their definition by recurrence it follows that the Chebyshev polynomials can be obtained as determinants of special tridiagonal matrices of size $k\; \backslash times\; k$:

$$T\_k(x)\; =\; \backslash det$$

\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},

and similarly for $U\_k$.

[[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 {{math|T₀}}, {{math|T₁}}, {{math|T₂}}, {{math|T₃}}, {{math|T₄}} and {{math|T₅}}.]]

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

$$\backslash begin\{align\}$$

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}

[[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 {{math|U₀}}, {{math|U₁}}, {{math|U₂}}, {{math|U₃}}, {{math|U₄}} and {{math|U₅}}.

Although not visible in the image, {{math|1=U_n(1) = n + 1}} and {{math|1=U_n(−1) = (n + 1)(−1)ⁿ}}.]]

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

$$\backslash begin\{align\}$$

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}

Image:ChebyshevExpansion.png, 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 orthonormal basis, so that a function in the same space can, on {{math|−1 ≤ x ≤ 1}}, be expressed via the expansion:{{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}}

$$f(x)\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; a\_n\; T\_n(x).$$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients {{math|a_n}} 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. These attributes include:

• The Chebyshev polynomials form a complete orthogonal system.
• The Chebyshev series converges to {{math|f(x)}} if the function is piecewise smooth and 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, 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 Chebysev basis.

Consider the Chebyshev expansion of {{math|log(1 + x)}}. One can express:

$$\backslash log(1+x)\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; a\_n\; T\_n(x)~.$$

One can find the coefficients {{math|a_n}} either through the application of an inner product or by the discrete orthogonality condition. For the inner product:

$$\backslash int\_\{-1\}^\{+1\}\backslash ,\backslash frac\{T\_m(x)\backslash ,\backslash log(1\; +\; x)\}\{\backslash sqrt\{1-x^2\}\}\backslash ,\backslash mathrm\{d\}x\; =\; \backslash sum\_\{n=0\}^\{\backslash infty\}a\_n\backslash int\_\{-1\}^\{+1\}\backslash frac\{T\_m(x)\backslash ,T\_n(x)\}\{\backslash sqrt\{1-x^2\}\}\backslash ,\backslash mathrm\{d\}x,$$

which gives:

$$a\_n\; =\; \backslash begin\{cases\}$$

-\log 2 & \text{ for }~ n = 0, \\

\frac{-2(-1)^n}{n} & \text{ for }~ n > 0.

\end{cases}

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:

$$a\_n\; \backslash approx\; \backslash frac\{\backslash ,2-\backslash delta\_\{0n\}\backslash ,\}\{N\}\backslash ,\backslash sum\_\{k=0\}^\{N-1\}T\_n(x\_k)\backslash ,\backslash log(1+x\_k),$$

where {{mvar|δ_ij}} is the Kronecker delta function and the {{mvar|x_k}} are the {{mvar|N}} Gauss–Chebyshev zeros of {{math|T_N(x)}}:

$$x\_k\; =\; \backslash cos\backslash left(\backslash frac\{\backslash pi\backslash left(k+\backslash tfrac\{1\}\{2\}\backslash right)\}\{N\}\backslash right)\; .$$

For any {{mvar|N}}, these approximate coefficients provide an exact approximation to the function at {{mvar|x_k}} 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_n}} very efficiently through the discrete cosine transform:

$$a\_n\; \backslash approx\; \backslash frac\{2-\backslash delta\_\{0n\}\}\{N\}\backslash sum\_\{k=0\}^\{N-1\}\backslash cos\backslash left(\backslash frac\{n\backslash pi\backslash left(\backslash ,k+\backslash tfrac\{1\}\{2\}\backslash right)\}\{N\}\backslash right)\backslash log(1+x\_k).$$

To provide another example:

$$\backslash begin\{align\}$$

\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}

The partial sums of:

$$f(x)\; =\; \backslash sum\_\{n\; =\; 0\}^\backslash infty\; a\_n\; T\_n(x)$$

are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients {{mvar|a_n}} 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 {{mvar|N}} coefficients of the {{math|(N − 1)}}st partial sum are usually obtained on the Chebyshev–Gauss–Lobatto{{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 }} 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:

$$x\_k\; =\; -\backslash cos\backslash left(\backslash frac\{k\; \backslash pi\}\{N\; -\; 1\}\backslash right);\; \backslash qquad\; k\; =\; 0,\; 1,\; \backslash dots,\; N\; -\; 1.$$

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:

$$p(x)\; =\; \backslash sum\_\{n=0\}^N\; a\_n\; T\_n(x).$$

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

Polynomials denoted $C\_n(x)$ and $S\_n(x)$ closely related to Chebyshev polynomials are sometimes used. They are defined by:{{sfn|Hochstrasser|1972|p=778}}

$$C\_n(x)\; =\; 2T\_n\backslash left(\backslash frac\{x\}\{2\}\backslash right),\backslash qquad\; S\_n(x)\; =\; U\_n\backslash left(\backslash frac\{x\}\{2\}\backslash right)$$

and satisfy:

$$C\_n(x)\; =\; S\_n(x)\; -\; S\_\{n-2\}(x).$$

A. F. Horadam called the polynomials $C\_n(x)$ Vieta–Lucas polynomials and denoted them $v\_n(x)$. He called the polynomials

$S\_n(x)$ Vieta–Fibonacci polynomials and denoted them {{nowrap|$V\_n(x)$.}}{{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}} Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII.{{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}} The Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of $i$ and a shift of index in the case of the latter, equal to Lucas and Fibonacci polynomials {{math|L_n}} and {{math|F_n}} of imaginary argument.

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

$$T\_n^*(x)\; =\; T\_n(2x-1),\backslash qquad\; U\_n^*(x)\; =\; U\_n(2x-1).$$

When the argument of the Chebyshev polynomial satisfies {{math|2x − 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."{{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}} The Chebyshev polynomials of the third kind are defined as:

$$V\_n(x)=\backslash frac\{\backslash cos\backslash left(\backslash left(n+\backslash frac\{1\}\{2\}\backslash right)\backslash theta\backslash right)\}\{\backslash cos\backslash left(\backslash frac\{\backslash theta\}\{2\}\backslash right)\}=\backslash sqrt\backslash frac\{2\}\{1+x\}T\_\{2n+1\}\backslash left(\backslash sqrt\backslash frac\{x+1\}\{2\}\backslash right)$$

and the Chebyshev polynomials of the fourth kind are defined as:

$$W\_n(x)=\backslash frac\{\backslash sin\backslash left(\backslash left(n+\backslash frac\{1\}\{2\}\backslash right)\backslash theta\backslash right)\}\{\backslash sin\backslash left(\backslash frac\{\backslash theta\}\{2\}\backslash right)\}=U\_\{2n\}\backslash left(\backslash sqrt\backslash frac\{x+1\}\{2\}\backslash right),$$

where $\backslash theta=\backslash arccos\; x$.{{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}}

They coincide with the Dirichlet kernel.

In the airfoil literature $V\_n(x)$ and $W\_n(x)$ are denoted $t\_n(x)$ and $u\_n(x)$. The polynomial families $T\_n(x)$, $U\_n(x)$, $V\_n(x)$, and $W\_n(x)$ are orthogonal with respect to the weights:

$$\backslash left(1-x^2\backslash right)^\{-1/2\},\backslash quad\backslash left(1-x^2\backslash right)^\{1/2\},\backslash quad(1-x)^\{-1/2\}(1+x)^\{1/2\},\backslash quad(1+x)^\{-1/2\}(1-x)^\{1/2\}$$

and are proportional to Jacobi polynomials $P\_n^\{(\backslash alpha,\backslash beta)\}(x)$ with:

$$(\backslash alpha,\backslash beta)=\backslash left(-\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\}\backslash right),\backslash quad(\backslash alpha,\backslash beta)=\backslash left(\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\}\backslash right),\backslash quad(\backslash alpha,\backslash beta)=\backslash left(-\backslash frac\{1\}\{2\},\backslash frac\{1\}\{2\}\backslash right),\backslash quad(\backslash alpha,\backslash beta)=\backslash left(\backslash frac\{1\}\{2\},-\backslash frac\{1\}\{2\}\backslash right).$$

All four families satisfy the recurrence $p\_n(x)=2xp\_\{n-1\}(x)-p\_\{n-2\}(x)$ with $p\_0(x)\; =\; 1$, where $p\_n\; =\; T\_n$, $U\_n$, $V\_n$, or $W\_n$, but they differ according to whether $p\_1(x)$ equals $x$, $2x$, $2x-1$, or {{nowrap|$2x+1$.}}

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.{{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}} 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.

$$P\_N\; =\; \backslash prod\_\{i=1\}^N(x-C\_i)$$

where

• $P\_N$ is an N-th order Chebyshev polynomial
• $C\_i$ is the i-th Chebyshev node

In the case of even order modified Chebyshev polynomials, the even order modified Chebyshev nodes are used to construct the even order modified Chebyshev polynomials.

$$Pe\_N\; =\; \backslash prod\_\{i=1\}^N(x-Ce\_i)$$

where

• $P\; e\_N$ is an N-th order even order modified Chebyshev polynomial
• $Ce\_i$ is the i-th even order modified Chebyshev node

For example, the 4th order Chebyshev polynomial from the example above is $X^4-X^2+.125$

, 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 $X^4-.828427X^2$

, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.

{{Portal|Mathematics}}

• Chebyshev rational functions
• Function approximation
• Discrete Chebyshev transform
• Markov brothers' inequality

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Category:Special hypergeometric functions

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