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
Full paper (PDF, 3562 Kb)

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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, T_n(x) = cos(n cos⁻¹ x) (n integer, |x| < 1), satisfy the second-order recurrence relation T_{n + 2} = 2xT_n+1 − T_n, T₀ = 1, T1 = x. It is shown that they also satisfy the first-order recurrence relation T_n+1 = xT_n + r((1 − x²)(1 − T_n²)), 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 X_n+2 = 2a(x)X_n+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 V_n(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.