Decomposition of Polynomials with Respect to the Cyclic Group of Orderm

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) = ∑kαj,kxλj, kcontaining only exponentsλj, 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.