Decomposition of perturbed Chebyshev polynomials

• Published in: Journal of computational and applied mathematics Vol. 214; no. 2; pp. 356 - 370
• Main Author: Stoll, Thomas
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.05.2008; Elsevier
• Subjects: Algebra; Chebyshev polynomials; Co-recursive and co-dilated orthogonal polynomials; Commutative rings and algebras; Exact sciences and technology; Fourier analysis; Gröbner bases; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Polynomial decomposition; Sciences and techniques of general use; Special functions; secondary; Gröbner bases; Co-recursive and co-dilated orthogonal polynomials; Polynomial decomposition; 30D05; Chebyshev polynomials; primary; Recurrence relation; Chebyshev polynomial; Decomposition method; Polynomial equation; Polynomial method; Algorithm; Orthogonal polynomial; primary 33C45; Equation system; Numerical analysis; secondary 33C47;30D05; Grobner bases; Applied mathematics
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2007.03.002

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.