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)**

## Details

### Authors and affiliations

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}= 2

*xT*−

_{n}_{+1}*T*,

_{n}*T*

_{0}= 1,

*T*1 =

*x*. It is shown that they also satisfy the first-order recurrence relation

*T*=

_{n}_{+1}*xT*+ r((1 −

_{n}*x*)(1 −

^{2}*T*

_{n}^{2})),

*T*

_{0}= 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}= 2

*a*(

*x*)

*X*

_{n}_{+1}−

*Xn*,

*X*

_{0}=

*x*

_{0},

*X*

_{1}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

- 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.

## Related papers

## 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.