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Chebyshev Polynomial of the First Kind

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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

T_n(z)=1/(4pii)∮((1-t^2)t^(-n-1))/((1-2tz+t^2))dt,
(1)

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

T_0(x)=1
(2)
T_1(x)=x
(3)
T_2(x)=2x^2-1
(4)
T_3(x)=4x^3-3x
(5)
T_4(x)=8x^4-8x^2+1
(6)
T_5(x)=16x^5-20x^3+5x
(7)
T_6(x)=32x^6-48x^4+18x^2-1.
(8)

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).

ChebyshevTSpiral

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

T_n(costheta)=cos(ntheta).
(9)

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

g_1(t,x)=(1-t^2)/(1-2xt+t^2)
(10)
=T_0(x)+2sum_(n=1)^(infty)T_n(x)t^n
(11)

and

g_2(t,x)=(1-xt)/(1-2xt+t^2)
(12)
=sum_(n=0)^(infty)T_n(x)t^n
(13)

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 is given by

T_n(z)=1/2z^2[(sqrt(1-1/(z^2))+1)^n+(sqrt(1-1/(z^2)))^n].
(14)

The polynomials can also be defined in terms of the sums

T_n(x)=n/2sum_(r=0)^(|_n/2_|)((-1)^r)/(n-r)(n-r; r)(2x)^(n-2r)
(15)
=cos(ncos^(-1)x)
(16)
=sum_(m=0)^(|_n/2_|)(n; 2m)x^(n-2m)(x^2-1)^m,
(17)

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

T_n(x)=2^(n-1)product_(k=1)^n{x-cos[((2k-1)pi)/(2n)]}
(18)

(Zwillinger 1995, p. 696).

T_n also satisfy the curious determinant equation

T_n=|x 1 0 0 ... 0 0; 1 2x 1 0 ... 0 0; 0 1 2x 1 ... 0 0; 0 0 1 2x ... 0 0; 0 0 0 1 ... 1 0; | ... ... ... ... ... 1; 0 0 0 0 ... 1 2x|
(19)

(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,

T_n(x)=(P_n^((-1/2,-1/2))(x))/(P_n^((-1/2,-1/2))(1))
(20)
=_2F_1(-n,n;1/2;1/2(1-x)),
(21)

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

Zeros occur when

x=cos[(pi(k-1/2))/n]
(22)

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

x=cos((pik)/n),
(23)

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)

int_(-1)^1(T_m(x)T_n(x)dx)/(sqrt(1-x^2))={1/2pidelta_(nm) for m!=0, n!=0; pi for m=n=0,
(24)

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

sum_(k=1)^mT_i(x_k)T_j(x_k)={1/2mdelta_(ij) for i!=0, j!=0; m for i=j=0,
(25)

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

They also satisfy the recurrence relations

T_(n+1)(x)=2xT_n(x)-T_(n-1)(x)
(26)
T_(n+1)(x)=xT_n(x)-sqrt((1-x^2){1-[T_n(x)]^2})
(27)

for n>=1, as well as

(x-1)[T_(2n+1)(x)-1]=[T_(n+1)(x)-T_n(x)]^2
(28)
2(x^2-1)[T_(2n)(x)-1]=[T_(n+1)(x)-T_(n-1)(x)]^2
(29)

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

They have a complex integral representation

T_n(x)=1/(4pii)int_gamma((1-z^2)z^(-n-1)dz)/(1-2xz+z^2)
(30)

and a Rodrigues representation

T_n(x)=((-1)^nsqrt(pi)(1-x^2)^(1/2))/(2^n(n-1/2)!)(d^n)/(dx^n)[(1-x^2)^(n-1/2)].
(31)

Using a fast Fibonacci transform with multiplication law

(A,B)(C,D)=(AD+BC+2xAC,BD-AC)
(32)

gives

(T_(n+1)(x),-T_n(x))=(T_1(x),-T_0(x))(1,0)^n.
(33)

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

p_0(x)=1
(34)
p_1(x)=[x-(int_(-1)^1x(1-x^2)^(-1/2)dx)/(int_(-1)^1(1-x^2)^(-1/2)dx)]
(35)
=x-([-(1-x^2)^(1/2)]_(-1)^1)/([sin^(-1)x]_(-1)^1)
(36)
=x
(37)
p_2(x)=[x-(int_(-1)^1x^3(1-x^2)^(-1/2)dx)/(int_(-1)^1x^2(1-x^2)^(-1/2)dx)]x-[(int_(-1)^1x^2(1-x^2)^(-1/2)dx)/(int_(-1)^1(1-x^2)^(-1/2)dx)]·1
(38)
=[x-0]x-(pi/2)/pi
(39)
=x^2-1/2,
(40)

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

J_n(x)=i^nT_n(id/(dx))J_0(x)
(41)
I_n(x)=T_n(d/(dx))I_0(x).
(42)

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

T_n(x)=cos(ntheta)
(43)
=cos(ncos^(-1)x).
(44)

The second linearly dependent solution to the transformed differential equation

(d^2T_n)/(dtheta^2)+n^2T_n=0
(45)

is then given by

V_n(x)=sin(ntheta)
(46)
=sin(ncos^(-1)x),
(47)

which can also be written

V_n(x)=sqrt(1-x^2)U_(n-1)(x),
(48)

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).

ChebyshevTPowers

The polynomials

p_n(x)=x^n-2^(1-n)T_n(x)
(49)

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

p_1(x)=0
(50)
p_2(x)=1/2
(51)
p_3(x)=3/4x
(52)
p_4(x)=x^2-1/8
(53)
p_5(x)=5/(16)(4x^3-x)
(54)

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

x=cos((kpi)/n),
(55)

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

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