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The **Chebyshev polynomials** are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions:

- The
*Chebyshev polynomials of the first kind*are given by

- Similarly, define the
*Chebyshev polynomials of the second kind*as

These definitions do not appear to be polynomials, but by using various trigonometric identities they can be converted to an explicitly polynomial form. For example, for *n* = 2 the *T*_{2} formula can be converted into a polynomial with argument *x* = cos(*θ*), using the double angle formula:

Replacing the terms in the formula with the definitions above, we get

The other *T _{n}*(

gives

Once converted to polynomial form, *T _{n}*(

Conversely, an arbitrary integer power of trigonometric functions may be expressed as a linear combination of trigonometric functions using Chebyshev polynomials

where the prime at the sum symbol indicates that the contribution of *j* = 0 needs to be halved if it appears, and .

An important and convenient property of the *T _{n}*(

and *U*_{n}(*x*) are orthogonal with respect to another, analogous inner product, given below. This follows from the fact that the Chebyshev polynomials solve the Chebyshev differential equations

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 Chebyshev polynomials *T _{n}* are polynomials with the largest possible leading coefficient, whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties.

Chebyshev polynomials are important in approximation theory because the roots of *T _{n}*(

These polynomials were named after Pafnuty Chebyshev.^{[2]} The letter T is used because of the alternative transliterations of the name *Chebyshev* as *Tchebycheff*, *Tchebyshev* (French) or *Tschebyschow* (German).

The **Chebyshev polynomials of the first kind** are obtained from the recurrence relation

The ordinary generating function for T_{n} is

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

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

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

Notice that the two sets of recurrence relations are identical, except for vs. .
The ordinary generating function for U_{n} is

the exponential generating function is

As described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying

or, in other words, as the unique polynomials satisfying

for *n* = 0, 1, 2, 3, … which as a technical point is a variant (equivalent transpose) of Schröder's equation. That is, *T _{n}*(

The polynomials of the second kind satisfy:

or

which is structurally quite similar to the Dirichlet kernel *D _{n}*(

That cos *nx* is an nth-degree polynomial in cos *x* can be seen by observing that cos *nx* is the real part of one side of de Moivre's formula. 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 *nx* is the imaginary part of the polynomial, in which all powers of sin *x* are odd and thus, if one is factored out, the remaining can be replaced to create a (*n*−1)th-degree polynomial in cos *x*.

The identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle.

Evaluating the first two Chebyshev polynomials,

and

one can straightforwardly determine that

and so forth.

Two immediate corollaries are the *composition identity* (or **nesting property** specifying a semigroup)

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

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

in a ring *R*[*x*].^{[3]} Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

The Chebyshev polynomials of the first and second kinds correspond to a complementary pair of Lucas sequences *Ṽ _{n}*(

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

The Chebyshev polynomials of the first and second kinds are also connected by the following relations:

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

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

Turán's inequalities for the Chebyshev polynomials are

The integral relations are

where integrals are considered as principal value.

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

with inverse^{[4]}^{[5]}

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

where _{2}*F*_{1} is a hypergeometric function.

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

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

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

Similarly, the roots of U_{n} are

The extrema of T_{n} on the interval −1 ≤ *x* ≤ 1 are located at

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

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

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

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

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

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

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

The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e. *U*_{n − 1}(1) = *nT _{n}*(1) =

The proof for *x* = −1 is similar, with the fact that *T _{n}*(−1) = (−1)

More general formula states:

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

Also, we have

where the prime at the summation symbols means that the term contributed by *k* = 0 is to be halved, if it appears.

Concerning integration, the first derivative of the T_{n} implies that

and the recurrence relation for the first kind polynomials involving derivatives establishes that for *n* ≥ 2

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:

Furthermore, we have

When working with Chebyshev polynomials quite often products of two of them occur. These products can be reduced to combinations of Chebyshev polynomials with lower or higher degree and concluding statements about the product are easier to make. It shall be assumed that in the following the index m is greater than or equal to the index n and n is not negative. For Chebyshev polynomials of the first kind the product expands to

which is an analogy to the addition theorem

with the identities

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 Chebyshev polynomials (depending on the parity of the lowest m) which allows to design functions with prescribed symmetry properties. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:

For Chebyshev polynomials of the second kind, products may be written as:

for *m* ≥ *n*.

By this, like above, with *n* = 2 the recurrence formula for Chebyshev polynomials of the second kind reduces for both types of symmetry to

depending on whether m starts with 2 or 3.

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

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

This can be proven by letting *x* = cos *θ* and using the defining identity *T*_{n}(cos *θ*) = cos *nθ*.

Similarly, the polynomials of the second kind U_{n} are orthogonal with respect to the weight

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

(The measure √1 − *x*^{2} d*x* is, to within a normalizing constant, the Wigner semicircle distribution.)

The T_{n} also satisfy a discrete orthogonality condition:

where N is any integer greater than max(*i*,*j*),^{[6]} and the *x*_{k} are the N Chebyshev nodes (see above) of *T*_{N}(*x*):

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

and without the weight function:

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

one can get the sum:

and again without the weight function:

For any given *n* ≥ 1, among the polynomials of degree n with leading coefficient 1 (*monic* polynomials),

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

This maximal absolute value is

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

Let's assume that *w _{n}*(

Define

Because at extreme points of T_{n} we have

From the intermediate value theorem, *f _{n}*(

By the Equioscillation theorem, among all the polynomials of degree ≤ *n*, the polynomial f minimizes ||*f*||_{∞} on [−1, 1] if and only if there are *n* + 2 points −1 ≤ *x*_{0} < *x*_{1} < ⋯ < *x*_{n + 1} ≤ 1 such that |*f*(*x _{i}*)| = ||

Of course, the null polynomial on the interval [−1,1] can be approach by itself and minimizes the ∞-norm.

Above, however, |*f*| reaches its maximum only *n* + 1 times because we are searching for the best polynomial of degree *n* ≥ 1 (therefore the theorem evoked previously cannot be used).

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

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

and

The leading coefficient of *T*_{n} is 2^{n − 1} if 1 ≤ *n*, but 1 if 0 = *n*.

*T*_{n} are a special case of Lissajous curves with frequency ratio equal to n.

Several polynomial sequences like Lucas polynomials (*L*_{n}), Dickson polynomials (*D*_{n}), Fibonacci polynomials (*F*_{n}) are related to Chebyshev polynomials *T*_{n} and *U*_{n}.

The Chebyshev polynomials of the first kind satisfy the relation

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

(with the definition *U*_{−1} ≡ 0 by convention ).

Similar to the formula

we have the analogous formula

For *x* ≠ 0,

and

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

Define

Then *C _{n}*(

as is evident in the Abelian nesting property specified above.

The generalized Chebyshev polynomials *T _{a}* are defined by

where a is not necessarily an integer, and _{2}*F*_{1}(*a*, *b*; *c*; *z*) is the Gaussian hypergeometric function; as an example, .
The power series expansion

converges for .

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

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

In the appropriate Sobolev space, the set of Chebyshev polynomials form an orthonormal basis, so that a function in the same space can, on −1 ≤ *x* ≤ 1 be expressed via the expansion:^{[7]}

Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients *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.^{[7]} These attributes include:

- The Chebyshev polynomials form a complete orthogonal system.
- The Chebyshev series converges to
*f*(*x*) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in*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,^{[7]} often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).

Consider the Chebyshev expansion of log(1 + *x*). One can express

One can find the coefficients *a _{n}* either through the application of an inner product or by the discrete orthogonality condition. For the inner product,

which gives

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,

where δ_{ij} is the Kronecker delta function and the x_{k} are the N Gauss–Chebyshev zeros of *T _{N}*(

For any N, these approximate coefficients provide an exact approximation to the function at x_{k} with a controlled error between those points. The exact coefficients are obtained with *N* = ∞, thus representing the function exactly at all points in [−1,1]. The rate of convergence depends on the function and its smoothness.

This allows us to compute the approximate coefficients a_{n} very efficiently through the discrete cosine transform

To provide another example:

The partial sums of

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 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 N coefficients of the (*N* − 1)th partial sum are usually obtained on the Chebyshev–Gauss–Lobatto^{[8]} 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:

An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.^{[9]} Such a polynomial *p*(*x*) is of the form

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

Shifted Chebyshev polynomials of the first kind are defined as

When the argument of the Chebyshev polynomial is in the range of 2*x* − 1 ∈ [−1, 1] the argument of the shifted Chebyshev polynomial is *x* ∈ [0, 1]. Similarly, one can define shifted polynomials for generic intervals [*a*,*b*].

**^**Rivlin, Theodore J. (1974). "Chapter 2, Extremal properties".*The Chebyshev Polynomials*. Pure and Applied Mathematics (1st ed.). New York-London-Sydney: Wiley-Interscience [John Wiley & Sons]. pp. 56–123. ISBN 978-047172470-4.**^**Chebyshev polynomials were first presented in Chebyshev, P. L. (1854). "Théorie des mécanismes connus sous le nom de parallélogrammes".*Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg*(in French).**7**: 539–586.**^**Demeyer, Jeroen (2007).*Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields*(PDF) (Ph.D. thesis). p. 70. Archived from the original (PDF) on 2 July 2007.**^**Cody, W. J. (1970). "A survey of practical rational and polynomial approximation of functions".*SIAM Review*.**12**(3): 400–423. doi:10.1137/1012082.**^**Mathar, R. J. (2006). "Chebyshev series expansion of inverse polynomials".*J. Comput. Appl. Math*.**196**(2): 596–607. arXiv:math/0403344. Bibcode:2006JCoAM..196.596M. doi:10.1016/j.cam.2005.10.013. S2CID 16476052.**^**For proof, see: Mason, J.C. & Handscomb, D.C. (2002).*Chebyshev Polynomials*. Taylor & Francis.- ^
^{a}^{b}^{c}Boyd, John P. (2001).*Chebyshev and Fourier Spectral Methods*(PDF) (second ed.). Dover. ISBN 0-486-41183-4. Archived from the original (PDF) on 31 March 2010. Retrieved 19 March 2009. **^**"Chebyshev Interpolation: An Interactive Tour". Archived from the original on 18 March 2017. Retrieved 2 June 2016.**^**For more information on the coefficients, see: Mason, J.C. & Handscomb, D.C. (2002).*Chebyshev Polynomials*. Taylor & Francis.

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- Media related to Chebyshev polynomials at Wikimedia Commons
- Weisstein, Eric W. "Chebyshev polynomial[s] of the first kind".
*MathWorld*. - Mathews, John H. (2003). "Module for Chebyshev polynomials". Department of Mathematics. Course notes for Math 340
*Numerical Analysis*& Math 440*Advanced Numerical Analysis*. Fullerton, CA: California State University. Archived from the original on 29 May 2007. Retrieved 17 August 2020. - "Chebyshev interpolation: An interactive tour". Mathematical Association of America (MAA) – includes illustrative Java applet.
- "Numerical computing with functions".
*The Chebfun Project*. - "Is there an intuitive explanation for an extremal property of Chebyshev polynomials?".
*Math Overflow*. Question 25534. - "Chebyshev polynomial evaluation and the Chebyshev transform".
*Boost*. Math.