Summary: 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 \(T_n\) and \(U_n\). We show that they are in a \(\mathbb{Q}\)-vectorial subspace \(E_n(x)\) of \(\mathbb{Q}[x]\) of dimension \(n\). We establish that the polynomial sequences \((h^kT_{n-k})_k\) and \((h^kU_{n-k})_k\), \((0\le k\le n-1)\) are two bases of \(E_n (x)\) for which \(T_n\) and \(U_n\) admit remarkable integer coordinates.