On some properties of Chebyshev polynomials

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Title

Authors ¹, ¹

Letting

T_{n}

(resp.

U_{n}

) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences

{(X^{k} T_{n - k})}_{k}

and

{(X^{k} U_{n - k})}_{k}

for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space

_{n} [X]

formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also

T_{n}

and

U_{n}

admit remarkableness integer coordinates on each of the two basis.

Chebyshev polynomials, integer coordinates

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E. Lucas, Théorie des Nombres, Ghautier-Villars, Paris 1891.
T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, second edition, Wiley Interscience 1990.