Chebyshev Polynomials

The purpose, use and mathematics of Chebyshev Polynomials are all well out of the scope of this project. Nonetheless, the important point to note in for this paper, is that at the exact same time the Gibonacci Sequence research was being conducted for the Theory of Order, Professor Milan Janjic of the Faculty of Natural Sciences and Mathematics, Banja Luka, located in Republic of Serbia was developing a combinatoric function to output the roots of Chebyshev polynomials.

When I asked Dr. Janjic for a general formula to produce what he called the roots of Chebyshev polynomials, he responded that I need wait two weeks while he finished his derivation. I did wait and in due time, he did respond.

Though as of yet unpublished, Dr. Janjic provided us with his solution for the development of a non-recursive function that produces Gibonacci sequences. The work, help and generosity of Dr. Janjic are greatly appreciated.


The function provided by Dr. Janjic is as follows



To try to put this in context, below is the Wolfram Mathematica entry for Chebyshev Polynomials of the first order:


From :

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 Mathematica 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

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

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, ... (Sloane's 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


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)
= T_0(x)+2sum_(n=1)^(infty)T_n(x)t^n


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

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


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

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

 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|

(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))
= _2F_1(-n,n;1/2;1/2(1-x)),

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)

 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,

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,

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

for n>=1, as well as

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

(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




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

p_0(x) = 1
p_1(x) = [x-(int_(-1)^1x(1-x^2)^(-1/2)dx)/(int_(-1)^1(1-x^2)^(-1/2)dx)]
= x-([-(1-x^2)^(1/2)]_(-1)^1)/([sin^(-1)x]_(-1)^1)
= x
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
= [x-0]x-(pi/2)/pi
= x^2-1/2,

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

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

The second linearly dependent solution to the transformed differential equation


is then given by

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

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}, ... (Sloane's A054375).


The polynomials


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

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

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).

SEE ALSO: Chebyshev Approximation Formula, Chebyshev Polynomial of the Second Kind



Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. "Chebyshev (Tschebyscheff) Polynomials" and "Chebyshev Polynomials--Numerical Applications." §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731-748, 1985.

Beeler et al. Item 15 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 9, Feb. 1972.

Iyanaga, S. and Kawada, Y. (Eds.). "Čebyšev (Tschebyscheff) Polynomials." Appendix A, Table 20.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1478-1479, 1980.

Koekoek, R. and Swarttouw, R. F. "Chebyshev." §1.8.2 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 41-43, 1998.

Koepf, W. "Efficient Computation of Chebyshev Polynomials." In Computer Algebra Systems: A Practical Guide (Ed. M. J. Wester). New York: Wiley, pp. 79-99, 1999.

Nash, P. L. "Chebyshev Polynomials and Quadratic Path Integrals." J. Math. Phys. 27, 2963, 1986.

Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990.

Shohat, J. Théorie générale des polynomes orthogonaux de Tchebichef. Paris: Gauthier-Villars, 1934.

Sloane, N. J. A. Sequences A008310 and A054375 in "The On-Line Encyclopedia of Integer Sequences."

Spanier, J. and Oldham, K. B. "The Chebyshev Polynomials T_n(x) and U_n(x)." Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987.

Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 10 and 84, 1999.

Vasilyev, N. and Zelevinsky, A. "A Chebyshev Polyplayground: Recurrence Relations Applied to a Famous Set of Formulas." Quantum 10, 20-26, Sept./Oct. 1999.

Watkins, W. and Zeitlin, J. "The Minimal Polynomial of cos(2pi/n)." Amer. Math. Monthly 100, 471-474, 1993.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.


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