Chebyshev polynomials: Difference between revisions

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File:Chebyshev Polynomials of the First Kind.svg

Plot of the first five Template:Mvar Chebyshev polynomials (first kind)

File:Chebyshev Polynomials of the Second Kind.svg

Plot of the first five Template:Mvar Chebyshev polynomials (second kind)

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as $T_{n} (x)$ and $U_{n} (x)$ . They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The Chebyshev polynomials of the first kind $T_{n}$ are defined by

$T_{n} (\cos θ) = \cos (n θ) .$

Similarly, the Chebyshev polynomials of the second kind $U_{n}$ are defined by

$U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ) .$

That these expressions define polynomials in $\cos θ$ is not obvious at first sight but can be shown using de Moivre's formula (see below).

The Chebyshev polynomials Template:Math are polynomials with the largest possible leading coefficient whose absolute value on the interval Template:Closed-closed is bounded by 1. They are also the "extremal" polynomials for many other properties.^[1]

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;^[2] the roots of Template:Math, 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.^[3] The letter Template:Mvar is used because of the alternative transliterations of the name Chebyshev as Script error: No such module "Lang"., Script error: No such module "Lang". (French) or Script error: No such module "Lang". (German).

Definitions

Recurrence definition

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

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

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

$\begin{aligned} U_{0} (x) & = 1, \\ U_{1} (x) & = 2 x, \\ U_{n + 1} (x) & = 2 x U_{n} (x) - U_{n - 1} (x), \end{aligned}$ 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

$T_{n} (\cos θ) = \cos (n θ)$

and

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

for Template:Math.

An equivalent way to state this is via exponentiation of a complex number: given a complex number Template:Math with absolute value of one,

$z^{n} = T_{n} (a) + i b U_{n - 1} (a) .$

Chebyshev polynomials can be defined in this form when studying trigonometric polynomials.^[4]

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

$\cos n θ + i \sin n θ = (\cos θ + i \sin θ)^{n} .$

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

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

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

Commuting polynomials definition

Chebyshev polynomials can also be characterized by the following theorem:^[5]

If $F_{n} (x)$ is a family of monic polynomials with coefficients in a field of characteristic $0$ such that $\deg F_{n} (x) = n$ 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 $F_{n} (x) = 2 \cdot T_{n} (x / 2)$ for all $n$ .

Pell equation definition

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

$T_{n} (x)^{2} - (x^{2} - 1) U_{n - 1} (x)^{2} = 1$

in a ring $R [x]$ .^[6] Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

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

Generating functions

The ordinary generating function for $T_{n}$ is

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

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

$\begin{aligned} \sum_{n = 0}^{\infty} T_{n} (x) \frac{t^{n}}{n!} & = \frac{1}{2} (\exp (t (x - \sqrt{x^{2} - 1})) + \exp (t (x + \sqrt{x^{2} - 1}))) \\ = e^{t x} \cosh (t \sqrt{x^{2} - 1}) . \end{aligned}$

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

$\sum_{n = 1}^{\infty} T_{n} (x) \frac{t^{n}}{n} = \ln (\frac{1}{\sqrt{1 - 2 t x + t^{2}}}) .$

The ordinary generating function for Template:Mvar is

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

and the exponential generating function is

$\sum_{n = 0}^{\infty} U_{n} (x) \frac{t^{n}}{n!} = e^{t x} (\cosh (t \sqrt{x^{2} - 1}) + \frac{x}{\sqrt{x^{2} - 1}} \sinh (t \sqrt{x^{2} - 1})) .$

Relations between the two kinds of Chebyshev polynomials

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

$\begin{aligned} {\tilde{U}}_{n} (2 x, 1) & = U_{n - 1} (x), \\ {\tilde{V}}_{n} (2 x, 1) & = 2 T_{n} (x) . \end{aligned}$

It follows that they also satisfy a pair of mutual recurrence equations:Template:Sfn

$\begin{aligned} T_{n + 1} (x) & = x T_{n} (x) - (1 - x^{2}) U_{n - 1} (x), \\ U_{n + 1} (x) & = x U_{n} (x) + T_{n + 1} (x) . \end{aligned}$

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

$T_{n} (x) = \frac{1}{2} (U_{n} (x) - U_{n - 2} (x)) .$

Using this formula iteratively gives the sum formula:

$U_{n} (x) = {\begin{cases} 2 \sum_{odd j > 0}^{n} T_{j} (x) & for odd n . \\ 2 \sum_{even j \geq 0}^{n} T_{j} (x) - 1 & for even n, \end{cases}$

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 $T_{n}$ :

$2 T_{n} (x) = \frac{1}{n + 1} \frac{d}{d x} T_{n + 1} (x) - \frac{1}{n - 1} \frac{d}{d x} T_{n - 1} (x), n = 2, 3, \dots$

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

Turán's inequalities for the Chebyshev polynomials are:^[7]

$\begin{aligned} T_{n} (x)^{2} - T_{n - 1} (x) T_{n + 1} (x) & = 1 - x^{2} > 0 & for - 1 < x < 1 & and \\ U_{n} (x)^{2} - U_{n - 1} (x) U_{n + 1} (x) & = 1 > 0 . \end{aligned}$

The integral relations areTemplate:Sfn Template:Sfn

$\begin{aligned} \int_{- 1}^{1} \frac{T_{n} (y)}{y - x} \frac{d y}{\sqrt{1 - y^{2}}} & = π U_{n - 1} (x), \\ \int_{- 1}^{1} \frac{U_{n - 1} (y)}{y - x} \sqrt{1 - y^{2}} d y & = - π T_{n} (x) \end{aligned}$

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 Template:Tmath:Script error: No such module "Unsubst".

$\begin{aligned} T_{n} (x) & = \frac{1}{2} ((x - \sqrt{x^{2} - 1})^{n} + (x + \sqrt{x^{2} - 1})^{n}) \\ = \frac{1}{2} ((x - \sqrt{x^{2} - 1})^{n} + (x - \sqrt{x^{2} - 1})^{- n}) . \end{aligned}$

The two are equivalent because $(x + \sqrt{x^{2} - 1}) (x - \sqrt{x^{2} - 1}) = 1$ .

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

$T_{n} (\cos (θ)) = Re (\cos n θ + i \sin n θ) = Re ((\cos θ + i \sin θ)^{n}),$

where $R e$ denotes the real part of a complex number. Expanding the formula, one gets

$(\cos θ + i \sin θ)^{n} = \sum_{j = 0}^{n} (\binom{n}{j}) i^{j} \sin^{j} θ \cos^{n - j} θ .$

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

$\cos n θ = \sum_{j = 0}^{⌊ n / 2 ⌋} (\binom{n}{2 j}) (\cos^{2} θ - 1)^{j} \cos^{n - 2 j} θ,$

which in turn means that

$T_{n} (x) = \sum_{j = 0}^{⌊ n / 2 ⌋} (\binom{n}{2 j}) (x^{2} - 1)^{j} x^{n - 2 j} .$

This can be written as a Template:Math hypergeometric function:

$\begin{aligned} T_{n} (x) & = \sum_{k = 0}^{⌊ \frac{n}{2} ⌋} (\binom{n}{2 k}) {(x^{2} - 1)}^{k} x^{n - 2 k} \\ = x^{n} \sum_{k = 0}^{⌊ \frac{n}{2} ⌋} (\binom{n}{2 k}) {(1 - x^{- 2})}^{k} \\ = \frac{n}{2} \sum_{k = 0}^{⌊ \frac{n}{2} ⌋} (- 1)^{k} \frac{(n - k - 1)!}{k! (n - 2 k)!} (2 x)^{n - 2 k} for n > 0 \\ = n \sum_{k = 0}^{n} (- 2)^{k} \frac{(n + k - 1)!}{(n - k)! (2 k)!} (1 - x)^{k} for n > 0 \\ =_{2} F_{1} (- n, n; \frac{1}{2}; \frac{1}{2} (1 - x)) \end{aligned}$

with inverse^[8]^[9]

$x^{n} = 2^{1 - n} {\sum^{'}}_{\binom{j = 0}{j \equiv n (\mod 2)}}^{n} (\binom{n}{\frac{n - j}{2}}) T_{j} (x),$

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

$T_{n} (x) = \sum_{m = 0}^{⌊ \frac{n}{2} ⌋} (- 1)^{m} ((\binom{n - m}{m}) + (\binom{n - m - 1}{n - 2 m})) \cdot 2^{n - 2 m - 1} \cdot x^{n - 2 m} .$

Similarly, $U_{n}$ can be expressed in terms of hypergeometric functions:

$\begin{aligned} U_{n} (x) & = \frac{{(x + \sqrt{x^{2} - 1})}^{n + 1} - {(x - \sqrt{x^{2} - 1})}^{n + 1}}{2 \sqrt{x^{2} - 1}} \\ = \sum_{k = 0}^{⌊ n / 2 ⌋} (\binom{n + 1}{2 k + 1}) {(x^{2} - 1)}^{k} x^{n - 2 k} \\ = x^{n} \sum_{k = 0}^{⌊ n / 2 ⌋} (\binom{n + 1}{2 k + 1}) {(1 - x^{- 2})}^{k} \\ = \sum_{k = 0}^{⌊ n / 2 ⌋} (\binom{2 k - (n + 1)}{k}) (2 x)^{n - 2 k} & for n > 0 \\ = \sum_{k = 0}^{⌊ n / 2 ⌋} (- 1)^{k} (\binom{n - k}{k}) (2 x)^{n - 2 k} & for n > 0 \\ = \sum_{k = 0}^{n} (- 2)^{k} \frac{(n + k + 1)!}{(n - k)! (2 k + 1)!} (1 - x)^{k} & for n > 0 \\ = (n + 1)_{2} F_{1} (- n, n + 2; \frac{3}{2}; \frac{1}{2} (1 - x)) . \end{aligned}$

Properties

Symmetry

$\begin{aligned} T_{n} (- x) & = (- 1)^{n} T_{n} (x), \\ U_{n} (- x) & = (- 1)^{n} U_{n} (x) . \end{aligned}$

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 $x$ .

Roots and extrema

A Chebyshev polynomial of either kind with degree Template:Mvar has Template:Mvar different simple roots, called Chebyshev roots, in the interval Template:Closed-closed. The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that:

$\cos ((2 k + 1) \frac{π}{2}) = 0$

one can show that the roots of $T_{n}$ are:

$x_{k} = \cos (\frac{π (k + 1 / 2)}{n}), k = 0, \dots, n - 1 .$

Similarly, the roots of $U_{n}$ are:

$x_{k} = \cos (\frac{k}{n + 1} π), k = 1, \dots, n .$

The extrema of $T_{n}$ on the interval $- 1 \leq x \leq 1$ are located at:

$x_{k} = \cos (\frac{k}{n} π), k = 0, \dots, n .$

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

$\begin{aligned} T_{n} (1) & = 1 \\ T_{n} (- 1) & = (- 1)^{n} \\ U_{n} (1) & = n + 1 \\ U_{n} (- 1) & = (- 1)^{n} (n + 1) . \end{aligned}$

The extrema of $T_{n} (x)$ on the interval $- 1 \leq x \leq 1$ where $n > 0$ are located at $n + 1$ values of $x$ . They are $\pm 1$ , or $\cos (\frac{2 π k}{d})$ where $d > 2$ , $d | 2 n$ , $0 < k < d / 2$ and $(k, d) = 1$ , i.e., $k$ and $d$ are relatively prime numbers.

Specifically (Minimal polynomial of 2cos(2pi/n)^[10]^[11]) when $n$ is even:

$T_{n} (x) = 1$ if $x = \pm 1$ , or $d > 2$ and $2 n / d$ is even. There are $n / 2 + 1$ such values of $x$ .
$T_{n} (x) = - 1$ if $d > 2$ and $2 n / d$ 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 $2 n / d$ is even. There are $(n + 1) / 2$ such values of $x$ .
$T_{n} (x) = - 1$ if $x = - 1$ , or $d > 2$ and $2 n / d$ is odd. There are $(n + 1) / 2$ such values of $x$ .

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:

$\begin{aligned} \frac{d T_{n}}{d x} & = n U_{n - 1} \\ \frac{d U_{n}}{d x} & = \frac{(n + 1) T_{n + 1} - x U_{n}}{x^{2} - 1} \\ \frac{d^{2} T_{n}}{d x^{2}} & = n \frac{n T_{n} - x U_{n - 1}}{x^{2} - 1} = n \frac{(n + 1) T_{n} - U_{n}}{x^{2} - 1} . \end{aligned}$

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

$\begin{aligned} {\frac{d^{2} T_{n}}{d x^{2}} |}_{x = 1} & = \frac{n^{4} - n^{2}}{3}, \\ {\frac{d^{2} T_{n}}{d x^{2}} |}_{x = - 1} & = (- 1)^{n} \frac{n^{4} - n^{2}}{3} . \end{aligned}$

More generally,

${\frac{d^{p} T_{n}}{d x^{p}} |}_{x = \pm 1} = (\pm 1)^{n + p} \prod_{k = 0}^{p - 1} \frac{n^{2} - k^{2}}{2 k + 1},$

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

Also, we have:

$\frac{d^{p}}{d x^{p}} T_{n} (x) = 2^{p} n {\sum^{'}}_{\binom{0 \leq k \leq n - p}{k \equiv n - p (\mod 2)}} (\binom{\frac{n + p - k}{2} - 1}{\frac{n - p - k}{2}}) \frac{(\frac{n + p + k}{2} - 1)!}{(\frac{n - p + k}{2})!} T_{k} (x), p \geq 1,$

where the prime at the summation symbols means that the term contributed by Template:Math is to be halved, if it appears.

Concerning integration, the first derivative of the Template:Mvar implies that:

$\int U_{n} d x = \frac{T_{n + 1}}{n + 1}$

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

$\int T_{n} d x = \frac{1}{2} (\frac{T_{n + 1}}{n + 1} - \frac{T_{n - 1}}{n - 1}) = \frac{n T_{n + 1}}{n^{2} - 1} - \frac{x T_{n}}{n - 1} .$

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:

$\begin{aligned} \int T_{n} d x & = \frac{n}{n^{2} - 1} T_{n + 1} - \frac{1}{n - 1} T_{1} T_{n} \\ = \frac{n}{n^{2} - 1} T_{n + 1} - \frac{1}{2 (n - 1)} (T_{n + 1} + T_{n - 1}) \\ = \frac{1}{2 (n + 1)} T_{n + 1} - \frac{1}{2 (n - 1)} T_{n - 1} . \end{aligned}$

Furthermore, we have:

$\int_{- 1}^{1} T_{n} (x) d x = {\begin{cases} \frac{(- 1)^{n} + 1}{1 - n^{2}} & if n \neq 1 \\ 0 & if n = 1 . \end{cases}$

Products of Chebyshev polynomials

The Chebyshev polynomials of the first kind satisfy the relation:

$T_{m} (x) T_{n} (x) = \frac{1}{2} (T_{m + n} (x) + T_{| m - n |} (x)), \forall m, n \geq 0,$

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

$2 \cos α \cos β = \cos (α + β) + \cos (α - β) .$

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 Template:Mvar) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:

$\begin{aligned} T_{2 n} (x) & = 2 T_{n}^{2} (x) - T_{0} (x) & = 2 T_{n}^{2} (x) - 1, \\ T_{2 n + 1} (x) & = 2 T_{n + 1} (x) T_{n} (x) - T_{1} (x) & = 2 T_{n + 1} (x) T_{n} (x) - x, \\ T_{2 n - 1} (x) & = 2 T_{n - 1} (x) T_{n} (x) - T_{1} (x) & = 2 T_{n - 1} (x) T_{n} (x) - x . \end{aligned}$

The polynomials of the second kind satisfy the similar relation:

$T_{m} (x) U_{n} (x) = {\begin{cases} \frac{1}{2} (U_{m + n} (x) + U_{n - m} (x)), & if n \geq m - 1, \\ \frac{1}{2} (U_{m + n} (x) - U_{m - n - 2} (x)), & if n \leq m - 2 . \end{cases}$

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

$U_{m} (x) U_{n} (x) = \sum_{k = 0}^{n} U_{m - n + 2 k} (x) = \sum_{\underset{step 2}{p = m - n}}^{m + n} U_{p} (x) .$

for $m \geq n$ . For $n = 2$ this recurrence reduces to:

$U_{m + 2} (x) = U_{2} (x) U_{m} (x) - U_{m} (x) - U_{m - 2} (x) = U_{m} (x) (U_{2} (x) - 1) - U_{m - 2} (x),$

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.

Composition and divisibility properties

The trigonometric definitions of $T_{n}$ and $U_{n}$ imply the composition or nesting properties:^[12]

$\begin{aligned} T_{m n} (x) & = T_{m} (T_{n} (x)), \\ U_{m n - 1} (x) & = U_{m - 1} (T_{n} (x)) U_{n - 1} (x) . \end{aligned}$

For $T_{m n}$ 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 $T_{m n} (x)$ is divisible by $T_{n} (x)$ if $m$ is odd. Furthermore, $U_{m n - 1} (x)$ is divisible by $U_{n - 1} (x)$ , and in the case that $m$ is even, divisible by $T_{n} (x) U_{n - 1} (x)$ .

Orthogonality

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:

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

on the interval Template:Closed-closed, i.e. we have:

$\int_{- 1}^{1} T_{n} (x) T_{m} (x) \frac{d x}{\sqrt{1 - x^{2}}} = {\begin{cases} 0 & if n \neq m, \\ π & if n = m = 0, \\ \frac{π}{2} & if n = m \neq 0 . \end{cases}$

This can be proven by letting $x = \cos (θ)$ and using the defining identity $T_{n} (\cos (θ) = \cos (n θ)$ .

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

$\sqrt{1 - x^{2}}$ on the interval Template:Closed-closed, i.e. we have:

$\int_{- 1}^{1} U_{n} (x) U_{m} (x) \sqrt{1 - x^{2}} d x = {\begin{cases} 0 & if n \neq m, \\ \frac{π}{2} & if n = m . \end{cases}$

(The measure $\sqrt{1 - x^{2}} d x$ 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:

$\begin{aligned} (1 - x^{2}) {T_{n}}^{″} - x {T_{n}}^{'} + n^{2} T_{n} & = 0, \\ (1 - x^{2}) {U_{n}}^{″} - 3 x {U_{n}}^{'} + n (n + 2) U_{n} & = 0, \end{aligned}$ 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:

$\sum_{k = 0}^{N - 1} T_{i} (x_{k}) T_{j} (x_{k}) = {\begin{cases} 0 & if i \neq j, \\ N & if i = j = 0, \\ \frac{N}{2} & if i = j \neq 0, \end{cases}$

where $N$ is any integer greater than $\max (i, j)$ ,Template:Sfn and the $x_{k}$ are the $N$ Chebyshev nodes (see above) of $T_{N} (x)$ :

$x_{k} = \cos (π \frac{2 k + 1}{2 N}) for k = 0, 1, \dots, N - 1 .$

For the polynomials of the second kind and any integer $N > i + j$ with the same Chebyshev nodes $x_{k}$ , there are similar sums:

$\sum_{k = 0}^{N - 1} U_{i} (x_{k}) U_{j} (x_{k}) (1 - x_{k}^{2}) = {\begin{cases} 0 & if i \neq j, \\ \frac{N}{2} & if i = j, \end{cases}$

and without the weight function:

$\sum_{k = 0}^{N - 1} U_{i} (x_{k}) U_{j} (x_{k}) = {\begin{cases} 0 & if i \equiv̸ j (\mod 2), \\ N \cdot (1 + \min {i, j}) & if i \equiv j (\mod 2) . \end{cases}$

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

$y_{k} = \cos (π \frac{k + 1}{N + 1}) for k = 0, 1, \dots, N - 1,$

one can get the sum:

$\sum_{k = 0}^{N - 1} U_{i} (y_{k}) U_{j} (y_{k}) (1 - y_{k}^{2}) = {\begin{cases} 0 & if i \neq j, \\ \frac{N + 1}{2} & if i = j, \end{cases}$

and again without the weight function:

$\sum_{k = 0}^{N - 1} U_{i} (y_{k}) U_{j} (y_{k}) = {\begin{cases} 0 & if i \equiv̸ j (\mod 2), \\ (\min {i, j} + 1) (N - \max {i, j}) & if i \equiv j (\mod 2) . \end{cases}$

Minimal Template:Math-norm

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

$f (x) = \frac{1}{2^{n - 1}} T_{n} (x)$

is the one of which the maximal absolute value on the interval Template:Closed-closed is minimal.

This maximal absolute value is:

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

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

$x = \cos \frac{k π}{n} for 0 \leq k \leq n .$

Template:Math proof

Remark

By the equioscillation theorem, among all the polynomials of degree Template:Math, the polynomial Template:Mvar minimizes Template:Math on Template:Closed-closed if and only if there are Template:Math points Template:Math such that Template:Math.

Of course, the null polynomial on the interval Template:Closed-closed can be approximated by itself and minimizes the Template:Math-norm.

Above, however, Template:Math reaches its maximum only Template:Math times because we are searching for the best polynomial of degree Template:Math (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 $C_{n}^{(λ)} (x)$ , which themselves are a special case of the Jacobi polynomials $P_{n}^{(α, β)} (x)$ :

$\begin{aligned} T_{n} (x) & = \frac{n}{2} \lim_{q \to 0} \frac{1}{q} C_{n}^{(q)} (x) if n \geq 1, \\ = \frac{1}{(\binom{n - \frac{1}{2}}{n})} P_{n}^{(- \frac{1}{2}, - \frac{1}{2})} (x) = \frac{2^{2 n}}{(\binom{2 n}{n})} P_{n}^{(- \frac{1}{2}, - \frac{1}{2})} (x), \\ U_{n} (x) & = C_{n}^{(1)} (x) \\ = \frac{n + 1}{(\binom{n + \frac{1}{2}}{n})} P_{n}^{(\frac{1}{2}, \frac{1}{2})} (x) = \frac{2^{2 n + 1}}{(\binom{2 n + 2}{n + 1})} P_{n}^{(\frac{1}{2}, \frac{1}{2})} (x) . \end{aligned}$

Chebyshev polynomials are also a special case of Dickson polynomials:

$D_{n} (2 x α, α^{2}) = 2 α^{n} T_{n} (x)$

$E_{n} (2 x α, α^{2}) = α^{n} U_{n} (x) .$

In particular, when $α = \frac{1}{2}$ , they are related by $D_{n} (x, \frac{1}{4}) = 2^{1 - n} T_{n} (x)$ and $E_{n} (x, \frac{1}{4}) = 2^{- n} U_{n} (x)$ .

Other properties

The curves given by Template:Math, or equivalently, by the parametric equations Template:Math, Template:Math, are a special case of Lissajous curves with frequency ratio equal to Template:Mvar.

Similar to the formula:

$T_{n} (\cos θ) = \cos (n θ),$

we have the analogous formula:

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

For Template:Math:

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

and:

$x^{n} = T_{n} (\frac{x + x^{- 1}}{2}) + \frac{x - x^{- 1}}{2} U_{n - 1} (\frac{x + x^{- 1}}{2}),$ which follows from the fact that this holds by definition for Template:Math.

There are relations between Legendre polynomials and Chebyshev polynomials

$\sum_{k = 0}^{n} P_{k} (x) T_{n - k} (x) = (n + 1) P_{n} (x)$

$\sum_{k = 0}^{n} P_{k} (x) P_{n - k} (x) = U_{n} (x)$

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 determinants of special tridiagonal matrices of size $k \times k$ :

$T_{k} (x) = \det [\begin{matrix} x & 1 & 0 & \dots & 0 \\ 1 & 2 x & 1 & ⋱ & ⋮ \\ 0 & 1 & 2 x & ⋱ & 0 \\ ⋮ & ⋱ & ⋱ & ⋱ & 1 \\ 0 & \dots & 0 & 1 & 2 x \end{matrix}],$ and similarly for $U_{k}$ .

Examples

First kind

File:Chebyshev Polynomials of the 1st Kind (n=0-5, x=(-1,1)).svg

The first few Chebyshev polynomials of the first kind in the domain Template:Math: The flat Template:Math, Template:Math, Template:Math, Template:Math, Template:Math and Template:Math.

The first few Chebyshev polynomials of the first kind are OEIS: A028297

$\begin{aligned} T_{0} (x) & = 1 \\ T_{1} (x) & = x \\ T_{2} (x) & = 2 x^{2} - 1 \\ T_{3} (x) & = 4 x^{3} - 3 x \\ T_{4} (x) & = 8 x^{4} - 8 x^{2} + 1 \\ T_{5} (x) & = 16 x^{5} - 20 x^{3} + 5 x \\ T_{6} (x) & = 32 x^{6} - 48 x^{4} + 18 x^{2} - 1 \\ T_{7} (x) & = 64 x^{7} - 112 x^{5} + 56 x^{3} - 7 x \\ T_{8} (x) & = 128 x^{8} - 256 x^{6} + 160 x^{4} - 32 x^{2} + 1 \\ T_{9} (x) & = 256 x^{9} - 576 x^{7} + 432 x^{5} - 120 x^{3} + 9 x \\ T_{10} (x) & = 512 x^{10} - 1280 x^{8} + 1120 x^{6} - 400 x^{4} + 50 x^{2} - 1 \end{aligned}$

Second kind

File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg

The first few Chebyshev polynomials of the second kind in the domain Template:Math: The flat Template:Math, Template:Math, Template:Math, Template:Math, Template:Math and Template:Math. Although not visible in the image, Template:Math and Template:Math.

The first few Chebyshev polynomials of the second kind are OEIS: A053117

$\begin{aligned} U_{0} (x) & = 1 \\ U_{1} (x) & = 2 x \\ U_{2} (x) & = 4 x^{2} - 1 \\ U_{3} (x) & = 8 x^{3} - 4 x \\ U_{4} (x) & = 16 x^{4} - 12 x^{2} + 1 \\ U_{5} (x) & = 32 x^{5} - 32 x^{3} + 6 x \\ U_{6} (x) & = 64 x^{6} - 80 x^{4} + 24 x^{2} - 1 \\ U_{7} (x) & = 128 x^{7} - 192 x^{5} + 80 x^{3} - 8 x \\ U_{8} (x) & = 256 x^{8} - 448 x^{6} + 240 x^{4} - 40 x^{2} + 1 \\ U_{9} (x) & = 512 x^{9} - 1024 x^{7} + 672 x^{5} - 160 x^{3} + 10 x \\ U_{10} (x) & = 1024 x^{10} - 2304 x^{8} + 1792 x^{6} - 560 x^{4} + 60 x^{2} - 1 \end{aligned}$

As a basis set

File:ChebyshevExpansion.png

The non-smooth function (top) Template:Math, where Template:Mvar 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 orthonormal basis, so that a function in the same space can, on Template:Math, be expressed via the expansion:^[13]

$f (x) = \sum_{n = 0}^{\infty} a_{n} T_{n} (x) .$

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients Template:Math 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.^[13] These attributes include:

The Chebyshev polynomials form a complete orthogonal system.
The Chebyshev series converges to Template:Math if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most casesTemplate:Snd as long as there are a finite number of discontinuities in Template:Math 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,^[13] 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.

Example 1

Consider the Chebyshev expansion of Template:Math. One can express:

$\log (1 + x) = \sum_{n = 0}^{\infty} a_{n} T_{n} (x) .$

One can find the coefficients Template:Math either through the application of an inner product or by the discrete orthogonality condition. For the inner product:

$\int_{- 1}^{+ 1} \frac{T_{m} (x) \log (1 + x)}{\sqrt{1 - x^{2}}} d x = \sum_{n = 0}^{\infty} a_{n} \int_{- 1}^{+ 1} \frac{T_{m} (x) T_{n} (x)}{\sqrt{1 - x^{2}}} d x,$ which gives: $a_{n} = {\begin{cases} - \log 2 & for n = 0, \\ \frac{- 2 (- 1)^{n}}{n} & for n > 0 . \end{cases}$

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:

$a_{n} \approx \frac{2 - δ_{0 n}}{N} \sum_{k = 0}^{N - 1} T_{n} (x_{k}) \log (1 + x_{k}),$

where Template:Mvar is the Kronecker delta function and the Template:Mvar are the Template:Mvar Gauss–Chebyshev zeros of Template:Math:

$x_{k} = \cos (\frac{π (k + \frac{1}{2})}{N}) .$

For any Template:Mvar, these approximate coefficients provide an exact approximation to the function at Template:Mvar with a controlled error between those points. The exact coefficients are obtained with Template:Math, thus representing the function exactly at all points in Template:Closed-closed. The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients Template:Mvar very efficiently through the discrete cosine transform:

$a_{n} \approx \frac{2 - δ_{0 n}}{N} \sum_{k = 0}^{N - 1} \cos (\frac{n π (k + \frac{1}{2})}{N}) \log (1 + x_{k}) .$

Example 2

To provide another example:

$\begin{aligned} {(1 - x^{2})}^{α} & = - \frac{1}{\sqrt{π}} \frac{Γ (\frac{1}{2} + α)}{Γ (α + 1)} + 2^{1 - 2 α} \sum_{n = 0} {(- 1)}^{n} (\binom{2 α}{α - n}) T_{2 n} (x) \\ = 2^{- 2 α} \sum_{n = 0} {(- 1)}^{n} (\binom{2 α + 1}{α - n}) U_{2 n} (x) . \end{aligned}$

Partial sums

The partial sums of:

$f (x) = \sum_{n = 0}^{\infty} a_{n} T_{n} (x)$

are very useful in the approximation of various functions and in the solution of differential equations (see spectral method). Two common methods for determining the coefficients Template:Mvar 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 Template:Mvar coefficients of the Template:Mathst partial sum are usually obtained on the Chebyshev–Gauss–Lobatto^[14] 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:

$x_{k} = - \cos (\frac{k π}{N - 1}); k = 0, 1, \dots, N - 1 .$

Polynomial in Chebyshev form

An arbitrary polynomial of degree Template:Mvar can be written in terms of the Chebyshev polynomials of the first kind.Template:Sfn Such a polynomial Template:Math is of the form:

$p (x) = \sum_{n = 0}^{N} a_{n} T_{n} (x) .$

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

Families of polynomials related to Chebyshev polynomials

Polynomials denoted $C_{n} (x)$ and $S_{n} (x)$ closely related to Chebyshev polynomials are sometimes used. They are defined by:Template:Sfn

$C_{n} (x) = 2 T_{n} (\frac{x}{2}), S_{n} (x) = U_{n} (\frac{x}{2})$

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 $V_{n} (x)$ .^[15] Lists of both sets of polynomials are given in Viète's Opera Mathematica, Chapter IX, Theorems VI and VII.^[16] 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 Template:Math and Template:Math of imaginary argument.

Shifted Chebyshev polynomials of the first and second kinds are related to the Chebyshev polynomials by:Template:Sfn

$T_{n}^{*} (x) = T_{n} (2 x - 1), U_{n}^{*} (x) = U_{n} (2 x - 1) .$

When the argument of the Chebyshev polynomial satisfies Template:Math the argument of the shifted Chebyshev polynomial satisfies Template:Math. Similarly, one can define shifted polynomials for generic intervals Template:Closed-closed.

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."^[17] The Chebyshev polynomials of the third kind are defined as:

$V_{n} (x) = \frac{\cos ((n + \frac{1}{2}) θ)}{\cos (\frac{θ}{2})} = \sqrt{\frac{2}{1 + x}} T_{2 n + 1} (\sqrt{\frac{x + 1}{2}})$ and the Chebyshev polynomials of the fourth kind are defined as: $W_{n} (x) = \frac{\sin ((n + \frac{1}{2}) θ)}{\sin (\frac{θ}{2})} = U_{2 n} (\sqrt{\frac{x + 1}{2}}),$

where $θ = \arccos x$ .^[17]^[18] 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:

${(1 - x^{2})}^{- 1 / 2}, {(1 - x^{2})}^{1 / 2}, (1 - x)^{- 1 / 2} (1 + x)^{1 / 2}, (1 + x)^{- 1 / 2} (1 - x)^{1 / 2}$

and are proportional to Jacobi polynomials $P_{n}^{(α, β)} (x)$ with:^[18]

$(α, β) = (- \frac{1}{2}, - \frac{1}{2}), (α, β) = (\frac{1}{2}, \frac{1}{2}), (α, β) = (- \frac{1}{2}, \frac{1}{2}), (α, β) = (\frac{1}{2}, - \frac{1}{2}) .$

All four families satisfy the recurrence $p_{n} (x) = 2 x p_{n - 1} (x) - p_{n - 2} (x)$ 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 $x$ , $2 x$ , $2 x - 1$ , or $2 x + 1$ .^[17]

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.^[19] 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.

$P_{N} = \prod_{i = 1}^{N} (x - C_{i})$

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.

$P e_{N} = \prod_{i = 1}^{N} (x - C e_{i})$

where

$P e_{N}$ is an N-th order even order modified Chebyshev polynomial
$C e_{i}$ 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 $X^{4} - . 828427 X^{2}$ , which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.

References

Template:Reflist

Sources

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External links

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Chebyshev polynomials: Difference between revisions

Revision as of 00:32, 20 June 2025

Contents

Definitions

Recurrence definition

Trigonometric definition

Commuting polynomials definition

Pell equation definition

Generating functions

Relations between the two kinds of Chebyshev polynomials

Explicit expressions

Properties

Symmetry

Roots and extrema

Differentiation and integration

Products of Chebyshev polynomials

Composition and divisibility properties

Orthogonality

Minimal Template:Math-norm

Remark

Chebyshev polynomials as special cases of more general polynomial families

Other properties

Chebyshev polynomials as determinants

Examples

First kind

Second kind

As a basis set

Example 1

Example 2

Partial sums

Polynomial in Chebyshev form

Families of polynomials related to Chebyshev polynomials

Even order modified Chebyshev polynomials

See also

References

Sources

Further reading

External links

Navigation menu

@@ Line 9: / Line 9: @@
 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>
@@ Line 27: / Line 29: @@
 The ''Chebyshev polynomials of the first kind'' can be defined by the recurrence relation
 <math display="block">\begin{align}
   T_0(x) & = 1, \\
@@ Line 34: / Line 37: @@
 The ''Chebyshev polynomials of the second kind'' can be defined by the recurrence relation
 <math display="block">\begin{align}
   U_0(x) & = 1, \\
@@ Line 44: / Line 48: @@
 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{{nnbsp}}''nx''}} is an {{mvar|n}}th-[[degree of a polynomial|degree]] polynomial in {{math|cos{{nnbsp}}''x''}} can be seen by observing that {{math|cos{{nnbsp}}''nx''}} is the [[complex number|real part]] of one side of [[de Moivre's formula]]:
+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{{nnbsp}}''x''}} and {{math|sin{{nnbsp}}''x''}}, in which all powers of {{math|sin{{nnbsp}}''x''}} are [[parity (mathematics)|even]] and thus replaceable through the identity {{math|1=cos<sup>2</sup>{{nnbsp}}''x'' + sin<sup>2</sup>{{nnbsp}}''x'' = 1}}. By the same reasoning, {{math|sin{{nnbsp}}''nx''}} is the [[complex number|imaginary part]] of the polynomial, in which all powers of {{math|sin{{nnbsp}}''x''}} are [[parity (mathematics)|odd]] and thus, if one factor of {{math|sin{{nnbsp}}''x''}} is factored out, the remaining factors can be replaced to create a {{math|(''n''&nbsp;−&nbsp;1)}}st-degree polynomial in {{math|cos{{nnbsp}}''x''}}.
-For ''x'' outside the interval [-1,1], the above definition implies
+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, \\
@@ Line 66: / Line 79: @@
 ===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}}</ref>
+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
@@ Line 74: / Line 87: @@
 ===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'']}}.<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:
+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 {{mvar|T<sub>n</sub>}} is
+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!}
@@ Line 92: / Line 110: @@
 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).
@@ Line 102: / Line 124: @@
 ==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|''Ṽ<sub>n</sub>''(''P'', ''Q'')}} and {{math|''Ũ<sub>n</sub>''(''P'', ''Q'')}} with parameters {{math|1=''P'' = 2''x''}} and {{math|1=''Q'' = 1}}:
+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), \\
@@ Line 114: / Line 139: @@
 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}
@@ Line 123: / Line 150: @@
 \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>:
@@ Line 130: / Line 158: @@
 [[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 }\\
@@ Line 136: / Line 165: @@
 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 expression:
+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"> T_n(x) = \dfrac{1}{2} \bigg( \Big(x-\sqrt{x^2-1} \Big)^n + \Big(x+\sqrt{x^2-1} \Big)^n \bigg) \quad \text{ for } x \in \mathbb {R},</math>
-<math display="block"> T_n(x) = \dfrac{1}{2} \bigg( \Big(x-\sqrt{x^2-1} \Big)^n + \Big(x-\sqrt{x^2-1} \Big)^{-n} \bigg) \quad \text{ for } x \in \mathbb {R}.</math>
+<math display="block">\begin{align}
-The two are equivalent because <math>(x + \sqrt{x^2 - 1})(x - \sqrt{x^2 - 1}) = 1</math>.
+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]]:
-An explicit form of the Chebyshev polynomial in terms of monomials {{math|''x''<sup>''k''</sup>}} 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|Re}} denotes the [[Complex number#Notation|real part]] of a complex number. Expanding the formula, one gets
+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} \\
@@ Line 167: / Line 212: @@
 & = {}_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.
@@ Line 176: / Line 222: @@
   | 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}} needs to be halved if it appears.
-A related expression for {{math|''T''<sub>''n''</sub>}} as a sum of monomials with binomial coefficients and powers of two is
+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''<sub>''n''</sub>}} can be expressed in terms of hypergeometric functions:
+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}} \\
@@ Line 203: / Line 253: @@
 \end{align}</math>
-That is, Chebyshev polynomials of even order have [[even and odd functions|even symmetry]] and therefore contain only even powers of {{mvar|x}}. Chebyshev polynomials of odd order have [[even and odd functions|odd symmetry]] and therefore contain only odd powers of {{mvar|x}}.
+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 {{mvar|T<sub>n</sub>}} are:
+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 {{mvar|U<sub>n</sub>}} are:
+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 {{mvar|T<sub>n</sub>}} on the interval {{math|−1 ≤ ''x'' ≤ 1}} are located at:
+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 ≤ ''x'' ≤ 1}} 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:
+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 \\
@@ Line 235: / Line 293: @@
 ===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} \\
@@ Line 241: / Line 300: @@
 \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|1=''x'' = 1}} and {{math|1=''x'' = −1}}. By [[L'Hôpital's rule]]:
+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}, \\
@@ Line 248: / Line 308: @@
 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'' ≥ 2}}:
+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 {{mvar|T<sub>n</sub>}} as a function of Chebyshev polynomials of the first kind only:
+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 \\
@@ Line 269: / Line 337: @@
 Furthermore, we have:
 <math display="block">\int_{-1}^1 T_n(x)\, \mathrm{d}x =
 \begin{cases}
@@ Line 277: / Line 346: @@
 ===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|1=''n'' = 1}} this results in the already known recurrence formula, just arranged differently, and with {{math|1=''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:
+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, \\
@@ Line 288: / Line 362: @@
 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,\\
@@ Line 293: / Line 368: @@
 \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''<sub>−1</sub> ≡ 0}} by convention ). They also satisfy:
+(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'' ≥ ''n''}}.
-For {{math|1=''n'' = 2}} this recurrence reduces to:
+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 {{mvar|m}} starts with 2 or 3.
+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''<sub>''n''</sub>}} and {{math|''U''<sub>''n''</sub>}} 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>
+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)),\\
@@ Line 307: / Line 388: @@
 \end{align}
 </math>
-For {{math|''T''<sub>''mn''</sub>}} the order of composition may be reversed, making the family of polynomial functions {{math|''T''<sub>''n''</sub>}} a [[commutative]] [[semigroup]] under composition.
-Since {{math|''T''<sub>''m''</sub>(''x'')}} is divisible by {{mvar|x}} if {{mvar|m}} is odd, it follows that {{math|''T''<sub>''mn''</sub>(''x'')}} is divisible by {{math|''T''<sub>''n''</sub>(''x'')}} if {{mvar|m}} is odd. Furthermore, {{math|''U''<sub>''mn''−1</sub>(''x'')}} is divisible by {{math|''U''<sub>''n''−1</sub>(''x'')}}, and in the case that {{mvar|m}} is even, divisible by {{math|''T''<sub>''n''</sub>(''x'')''U''<sub>''n''−1</sub>(''x'')}}.
+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 {{mvar|T<sub>n</sub>}} and {{mvar|U<sub>n</sub>}} form a sequence of [[orthogonal polynomials]]. The polynomials of the first kind {{mvar|T<sub>n</sub>}} are orthogonal with respect to the weight:
+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}
@@ Line 322: / Line 407: @@
 \end{cases}</math>
-This can be proven by letting {{math|1=''x'' = cos ''θ''}} and using the defining identity {{math|1=''T''<sub>''n''</sub>(cos ''θ'') = cos(''nθ'')}}.
+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}
@@ Line 333: / Line 420: @@
 \end{cases}</math>
-(The measure {{math|{{sqrt|1 − ''x''<sup>2</sup>}} d''x''}} is, to within a normalizing constant, the [[Wigner semicircle distribution]].)
+(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]].)
+\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:
-The {{mvar|T<sub>n</sub>}} also satisfy a discrete orthogonality condition:
 <math display="block">\sum_{k=0}^{N-1}{T_i(x_k)\,T_j(x_k)} =
 \begin{cases}
@@ Line 348: / Line 438: @@
 \frac{N}{2} & ~\text{ if }~ i = j \ne 0,
 \end{cases} </math>
-where {{mvar|N}} is any integer greater than {{math|max(''i'', ''j'')}},{{sfn|Mason|Handscomb|2002}} and the {{math|''x''<sub>''k''</sub>}} are the {{mvar|N}} [[Chebyshev nodes]] (see above) of {{math|''T''<sub>''N''&thinsp;</sub>(''x'')}}:
+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''}} with the same Chebyshev nodes {{math|''x''<sub>''k''</sub>}}, there are similar sums:
+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}
@@ Line 357: / Line 450: @@
 \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}
@@ Line 364: / Line 459: @@
 \end{cases} </math>
-For any integer {{math|''N'' > ''i'' + ''j''}}, based on the {{mvar|N}} zeros of {{math|''U''<sub>''N''&thinsp;</sub>(''x'')}}:
+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}
@@ Line 372: / Line 470: @@
 \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}
@@ Line 380: / Line 480: @@
 ===Minimal {{math|∞}}-norm===
-For any given {{math|''n'' ≥ 1}}, among the polynomials of degree {{mvar|n}} with leading coefficient 1 ([[monic polynomial|monic]] polynomials):
+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|{{abs|''f''(''x'')}}}} reaches this maximum exactly {{math|''n'' + 1}} times at:
+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<sub>n</sub>''(''x'')}} is a polynomial of degree {{mvar|n}} 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>}}.
+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| \\
@@ Line 414: / Line 521: @@
 ===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, \\
@@ Line 423: / Line 531: @@
 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>.
@@ Line 431: / Line 542: @@
 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>
@@ Line 453: / Line 570: @@
 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}
@@ Line 470: / Line 588: @@
 The first few Chebyshev polynomials of the first kind are {{OEIS2C|A028297}}
 <math display="block"> \begin{align}
 T_0(x) &= 1 \\
@@ Line 489: / Line 608: @@
 The first few Chebyshev polynomials of the second kind are {{OEIS2C|A053117}}
 <math display="block">\begin{align}
 U_0(x) &= 1 \\
@@ Line 507: / Line 627: @@
 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>
@@ Line 523: / Line 644: @@
 ===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:
@@ Line 534: / Line 657: @@
 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]
@@ Line 551: / Line 680: @@
 ===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>
@@ Line 565: / Line 698: @@
 == 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.
+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>
@@ Line 576: / Line 715: @@
 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>
@@ Line 591: / Line 735: @@
 === 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>
+<math display="block">P_N = \prod_{i=1}^N(x-C_i)</math>
 where
@@ Line 600: / Line 744: @@
 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>
+<math display="block">Pe_N = \prod_{i=1}^N(x-Ce_i)</math>
 where

Chebyshev polynomials: Difference between revisions

Revision as of 00:32, 20 June 2025

Definitions

Recurrence definition

Trigonometric definition

Commuting polynomials definition

Pell equation definition

Generating functions

Relations between the two kinds of Chebyshev polynomials

Explicit expressions

Properties

Symmetry

Roots and extrema

Differentiation and integration

Products of Chebyshev polynomials

Composition and divisibility properties

Orthogonality

Minimal Template:Math-norm

Remark

Chebyshev polynomials as special cases of more general polynomial families

Other properties

Chebyshev polynomials as determinants

Examples

First kind

Second kind

As a basis set

Example 1

Example 2

Partial sums

Polynomial in Chebyshev form

Families of polynomials related to Chebyshev polynomials

Even order modified Chebyshev polynomials

See also

References

Sources

Further reading

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