on Decomposition of Polynomials with Respect to the Cyclic Group of Order m †

Given a polynomial solution of a differential equation, its m-ary decomposition, i.e. its decomposition as a sum of m polynomials P [j] (x) = � k αj,kx λj,k containing only exponents λj,k with λj,k+1 − λj,k = m, is considered. A general algorithm is proposed in order to build holonomic equations for the m-ary parts P [j] (x) starting from the initial one, which, in addition, provides a factorized form of them. Moreover, these differential equations are used to compute expansions of the m-ary parts of a given polynomial in terms of classical orthogonal polynomials. As illustration, binary and ternary decomposition of these classical families are worked out in detail. c ○ 1999 Academic Press For each n = 0, 1,..., le