Chebyshev Polynomial of the First Kind

The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted T_n(x). They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer polynomial with alpha=0. They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the first kind are denoted T_n(x), and are implemented in the Wolfram Language as ChebyshevT[n, x]. They are normalized such that T_n(1)=1. The first few polynomials are illustrated above for x in [-1,1] and n=1, 2, ..., 5.

The Chebyshev polynomial of the first kind T_n(z) can be defined by the contour integral

where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).

The first few Chebyshev polynomials of the first kind are

When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 1; -1, 2; -3, 4; 1, -8, 8; 5, -20, 16, ... (OEIS A008310).

A beautiful plot can be obtained by plotting T_n(x) radially, increasing the radius for each value of n, and filling in the areas between the curves (Trott 1999, pp. 10 and 84).

The Chebyshev polynomials of the first kind are defined through the identity

or

The Chebyshev polynomials of the first kind can be obtained from the generating functions

and

for |x|<=1 and |t|<1 (Beeler et al. 1972, Item 15). (A closely related generating function is the basis for the definition of Chebyshev polynomial of the second kind.)

A direct representation in terms of powers of square roots is given by

The polynomials can also be defined in terms of the sums

where (n; k) is a binomial coefficient and |_x_| is the floor function, or the product

(Zwillinger 1995, p. 696).

T_n also satisfy the curious determinant equation

(Nash 1986).

The Chebyshev polynomials of the first kind are a special case of the Jacobi polynomials P_n^((alpha,beta)) with alpha=beta=-1/2,

where _2F_1(a,b;c;x) is a hypergeometric function (Koekoek and Swarttouw 1998).

Zeros occur when

for k=1, 2, ..., n. Extrema occur for

where k=0,1,...,n. At maximum, T_n(x)=1, and at minimum, T_n(x)=-1.

The Chebyshev polynomials are orthogonal polynomials with respect to the weighting function (1-x^2)^(-1/2)

where delta_(mn) is the Kronecker delta. Chebyshev polynomials of the first kind satisfy the additional discrete identity

where x_k for k=1, ..., m are the m zeros of T_m(x).

They also satisfy the recurrence relations

for n>=1, as well as

(Watkins and Zeitlin 1993; Rivlin 1990, p. 5).

They have a complex integral representation

and a Rodrigues representation

Using a fast Fibonacci transform with multiplication law

gives

Using Gram-Schmidt orthonormalization in the range (-1,1) with weighting function (1-x^2)^((-1/2)) gives

etc. Normalizing such that T_n(1)=1 gives the Chebyshev polynomials of the first kind.

The Chebyshev polynomial of the first kind is related to the Bessel function of the first kind J_n(x) and modified Bessel function of the first kind I_n(x) by the relations

Letting x=costheta allows the Chebyshev polynomials of the first kind to be written as

The second linearly dependent solution to the transformed differential equation

is then given by

which can also be written

where U_n is a Chebyshev polynomial of the second kind. Note that V_n(x) is therefore not a polynomial.

The triangle of resultants rho(T_n(x),T_k(x)) is given by {0}, {-1,0}, {0,-4,0}, {1,16,64,0}, {0,-16,0,4096,0}, ... (OEIS A054375).

The polynomials

of degree n-2, the first few of which are

are the polynomials of degree <n which stay closest to x^n in the interval (-1,1). The maximum deviation is 2^(1-n) at the n+1 points where

for k=0, 1, ..., n (Beeler et al. 1972).

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