First-order recurrence relations for the Chebyshev polynomials and associated function

Richard L. Ollerton and Richard N. Whitaker
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 4, 1998, Number 3, Pages 123—128
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Richard L. Ollerton
University of Western Sydney, Nepean 2774, Australia

Richard N. Whitaker
Bureau of Meteorology, Sydney, 2001, Australia

Abstract

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.

AMS Classification

  • 33C45
  • 11B37

References

  1. Abramowitz, M., and Stegun, I. A., 1970. Handbook of Mathematical Functions. Dover Publications, NY.
  2. 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.

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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.

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