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Generalized Chebyshev Polynomials

Abstract Let h(x) be a non constant polynomial with rational coefficients. Our aim is to introduce the h(x)-Chebyshev polynomials of the first and second kind Tn and Un. We show that they are in a ℚ-vectorial subspace En(x) of ℚ[x] of dimension n. We establish that the polynomial sequences (hkTn−k)k and (hkUn−k)k, (0 ≤ k ≤ n − 1) are two bases of En(x) for which Tn and Un admit remarkable integer coordinates.