Authors and affiliations
Richard L. Ollerton
University of Western Sydney, Nepean 2774, Australia
Richard N. Whitaker
Bureau of Meteorology, Sydney, 2001, Australia
The Chebyshev polynomials of the first kind, Tn(x) = cos(n cos−1 x) (n integer, |x| < 1), satisfy the second-order recurrence relation Tn + 2 = 2xTn+1 − Tn, T0 = 1, T1 = x. It is shown that they also satisfy the first-order recurrence relation Tn+1 = xTn + r((1 − x2)(1 − Tn2)), T0 = 1, where the function r is defined by r(p(x)2) = slc(p(x))p(x) for polynomial p(x) and slc(p(x)) denotes the sign of the leading coefficient of p(x).
Associated Chebyshev polynomials, satisfying Xn+2 = 2a(x)Xn+1 − Xn, X0 = x0, X1 polynomial, for polynomial a(x), are then defined and the corresponding first-order relation given. An example of non-polynomial a(x) leading to the functions Vn(x) = sin(n sin−1 x) is also mentioned together with a more general first-order recurrence relation for the non-polynomial case.
- Abramowitz, M., and Stegun, I. A., 1970. Handbook of Mathematical Functions. Dover Publications, NY.
- Horadam, A. F., 1998. New aspects of Morgan-Voyce polynomials. In: Applications of Fibonacci Funcions, Vol. 7, pp. 161-176. G. E. Bergum et al (eds), Kluwer Academic Publishers, Netherlands.
Cite this paper
Ollerton , R. L. & Whitaker, R. N. (1998). First-order recurrence relations for the Chebyshev polynomials and associated function. Notes on Number Theory and Discrete Mathematics, 4(3), 123-128.