Decomposition of perturbed Chebyshev polynomials

Abstract

We characterize polynomial decomposition f n = r ∘ q with r , q ∈ C [ x ] of perturbed Chebyshev polynomials defined by the recurrence f 0 ( x ) = b , f 1 ( x ) = x - c , f n + 1 ( x ) = ( x - d ) f n ( x ) - af n - 1 ( x ) , n ⩾ 1 , where a , b , c , d ∈ R and a > 0 . These polynomials generalize the Chebyshev polynomials, which are obtained by setting a = 1 4 , c = d = 0 and b ∈ { 1 , 2 } . At the core of the method, two algorithms for polynomial decomposition are provided, which allow to restrict the investigation to the resolution of six systems of polynomial equations in three variables. The final task is then carried out by the successful computation of reduced Gröbner bases with Maple 10. Some additional data for the calculations are available on the author's web page.