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Though as of yet unpublished, Dr.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 nonrecursive 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 : http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html 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 . They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multipleangle formulas. The Chebyshev polynomials of the first kind are denoted , and are implemented in Mathematica as ChebyshevT[n, x]. They are normalized such that . The first few polynomials are illustrated above for and , 2, ..., 5.
The Chebyshev polynomial of the first kind can be defined by the contour integral
(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
(2)


(3)


(4)


(5)


(6)


(7)


(8)

When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 1; , 2; , 4; 1, , 8; 5, , 16, ... (Sloane's A008310).
A beautiful plot can be obtained by plotting radially, increasing the radius for each value of , 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
(9)

The Chebyshev polynomials of the first kind can be obtained from the generating functions
(10)


(11)

and
(12)


(13)

for and (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
(14)

The polynomials can also be defined in terms of the sums
(15)


(16)


(17)

where is a binomial coefficient and is the floor function, or the product
(18)

(Zwillinger 1995, p. 696).
also satisfy the curious determinant equation
(19)

(Nash 1986).
The Chebyshev polynomials of the first kind are a special case of the Jacobi polynomials with ,
(20)


(21)

where is a hypergeometric function (Koekoek and Swarttouw 1998).
Zeros occur when
(22)

for , 2, ..., . Extrema occur for
(23)

where . At maximum, , and at minimum, .
The Chebyshev polynomials are orthogonal polynomials with respect to the weighting function
(24)

where is the Kronecker delta. Chebyshev polynomials of the first kind satisfy the additional discrete identity
(25)

where for , ..., are the zeros of .
They also satisfy the recurrence relations
(26)


(27)

for , as well as
(28)


(29)

(Watkins and Zeitlin 1993; Rivlin 1990, p. 5).
They have a complex integral representation
(30)

and a Rodrigues representation
(31)

Using a fast Fibonacci transform with multiplication law
(32)

gives
(33)

Using GramSchmidt orthonormalization in the range (,1) with weighting function gives
(34)


(35)


(36)


(37)


(38)


(39)


(40)

etc. Normalizing such that 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 and modified Bessel function of the first kind by the relations
(41)

(42)

Letting allows the Chebyshev polynomials of the first kind to be written as
(43)


(44)

The second linearly dependent solution to the transformed differential equation
(45)

is then given by
(46)


(47)

which can also be written
(48)

where is a Chebyshev polynomial of the second kind. Note that is therefore not a polynomial.
The triangle of resultants is given by , , , , , ... (Sloane's A054375).
The polynomials
(49)

of degree , the first few of which are
(50)


(51)


(52)


(53)


(54)

are the polynomials of degree which stay closest to in the interval . The maximum deviation is at the points where
(55)

for , 1, ..., (Beeler et al. 1972).
RELATED WOLFRAM SITES: http://functions.wolfram.com/Polynomials/ChebyshevT/, http://functions.wolfram.com/HypergeometricFunctions/ChebyshevTGeneral/
REFERENCES:
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. 771802, 1972.
Arfken, G. "Chebyshev (Tschebyscheff) Polynomials" and "Chebyshev PolynomialsNumerical Applications." §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731748, 1985.
Beeler et al. Item 15 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM239, p. 9, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/recurrence.html#item15.
Iyanaga, S. and Kawada, Y. (Eds.). "Čebyšev (Tschebyscheff) Polynomials." Appendix A, Table 20.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 14781479, 1980.
Koekoek, R. and Swarttouw, R. F. "Chebyshev." §1.8.2 in The AskeyScheme of Hypergeometric Orthogonal Polynomials and its Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 9817, pp. 4143, 1998.
Koepf, W. "Efficient Computation of Chebyshev Polynomials." In Computer Algebra Systems: A Practical Guide (Ed. M. J. Wester). New York: Wiley, pp. 7999, 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: GauthierVillars, 1934.
Sloane, N. J. A. Sequences A008310 and A054375 in "The OnLine Encyclopedia of Integer Sequences."
Spanier, J. and Oldham, K. B. "The Chebyshev Polynomials and ." Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193207, 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, 2026, Sept./Oct. 1999.
Watkins, W. and Zeitlin, J. "The Minimal Polynomial of ." Amer. Math. Monthly 100, 471474, 1993.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.
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