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Combinatorially composing Chebyshev polynomials

Authors:	Arthur T Benjamin Daniel Walton

Institution:	1. Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA;2. Department of Atmospheric and Oceanic Sciences, UCLA, Los Angeles, CA 90095-1565, USA

Abstract:	We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m,n?0 $m, n ? 0$ , _T_m(_T_n(x))=_T_mn(x) $T_{m} (T_{n} (x)) = T_{mn} (x)$ and _U_m₋₁(_T_n(x))_U_n₋₁(x)=_U_mn₋₁(x). $U_{m - 1} (T_{n} (x)) U_{n - 1} (x) = U_{mn - 1} (x) .$

Keywords:	Chebyshev polynomials Combinatorial proof Tiling
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