## Chebyshev Polynomial of the First Kind

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 Ultraspherical Polynomial with . The Chebyshev polynomials of the first kind are illustrated above for and , 2, ..., 5.

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

 (1)

and
 (2)

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.) They are normalized such that . They can also be written
 (3)

or in terms of a Determinant
 (4)

In closed form,
 (5)

where is a Binomial Coefficient and is the Floor Function. Therefore, zeros occur when
 (6)

for , 2, ..., . Extrema occur for
 (7)

where . At maximum, , and at minimum, . The Chebyshev Polynomials are Orthonormal with respect to the Weighting Function
 (8)

where is the Kronecker Delta. Chebyshev polynomials of the first kind satisfy the additional discrete identity
 (9)

where for , ..., are the zeros of . They also satisfy the Recurrence Relations
 (10)

 (11)

for . They have a Complex integral representation
 (12)

and a Rodrigues representation
 (13)

Using a Fast Fibonacci Transform with multiplication law
 (14)

gives
 (15)

Using Gram-Schmidt Orthonormalization in the range (,1) with Weighting Function gives

 (16) (17) (18)

etc. Normalizing such that gives

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

 (19)

 (20)

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

 (21)

The second linearly dependent solution to the transformed differential equation
 (22)

is then given by
 (23)

which can also be written
 (24)

where is a Chebyshev Polynomial of the Second Kind. Note that is therefore not a Polynomial.

The Polynomial

 (25)

(of degree ) is the Polynomial of degree which stays closest to in the interval . The maximum deviation is at the points where
 (26)

for , 1, ..., (Beeler et al. 1972, Item 15).

References

Abramowitz, M. and Stegun, C. 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, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

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

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

Spanier, J. and Oldham, K. B. The Chebyshev Polynomials and .'' Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987.