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Identities connecting the Chebyshev polynomials

Published online by Cambridge University Press:  17 October 2016

Jonny Griffiths*
20 Rosebery Road, Norwich NR3 3NA e-mail:


There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever nm. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.

Copyright © Mathematical Association 2016 

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1. Wikipedia, Fibonacci Polynomials, accessed February 2016 at Google Scholar
2. Wikipedia, Chebyshev Polynomials, accessed February 2016 at Google Scholar
3. Mason, J. C., Factorisation properties of Chebychev polynomials, Mathematics 49, (December 1993) pp. 169178.Google Scholar
4. The On-line Encyclopedia of Integer Sequences, accessed February 2016 Google Scholar
5. Griffiths, J., Lyness Cycles, Hikorski Triples and Elliptic Curves, MSc Thesis, University of East Anglia (2012), accessed February 2016, also available at Google Scholar
6. Rayes, M., Trevisan, V., and Wang, P., Factorisation properties of Chebychev polynomials, Comput. Math. Appl. 50 (2005) pp. 12311240.Google Scholar