The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas

Abstract. In this paper, the so-called compressed Chebyshev polynomials are studied, which are obtained from the original Chebyshev polynomials by suitable scaling transformations. By introducing the concept of cyclotomic pre-polynomials that are in one-to-one correspondence with the well known cyclotomic polynomials, explicit formulas for the irreducible factorization of all four kinds of compressed Chebyshev polynomials are developed. The correspondence between the cyclotomic pre-polynomials and cyclotomic polynomials is based on a polynomial transformation. The same transformation maps all four kinds of the compressed Chebyshev polynomials into polynomials having the property that all their roots are primitive roots of unity, and consequently they are easy to factorize. Generalizations of the applied methods, suitable for the factorization of similar classes of polynomials are also given in the paper.