On polynomial solutions of linear operator equations

The algorithm described here extends the algorithm to nd all polynomial solutions of di erential and di erence equations that was given in [1, 2] to more general operators. It also takes a more e cient approach that avoids using undetermined coe cients. This summary is based on [4]. Let K be a eld of characteristic 0 and L : K[x] ! K[x] a K-linear endomorphism of K[x]. A new algorithm is presented in [4] that nds all polynomial solutions of homogeneous equations of the form Ly = 0, of nonhomogeneous equations of the form Ly = f and of parametric nonhomogeneous equations of the form Ly = P m i=1 i f i . The endomorphisms L under consideration in the following are polynomials in one of the following operators, and with coe cients in K[x]: { the di erential operator D de ned by Df(x) = df=dx; { the di erence operator de ned by f(x) = f(x+ 1) f(x); { the q-dilation operator Q used for q-di erence equations and de ned by Qf(x) = f(qx). (In this case, q 2 K, is not zero and not a root of unity.) The interest of the new algorithm is twofold. First, numerous algorithms need to solve homogeneous, nonhomogeneous or parametric nonhomogeneous equations in K[x] as subproblems. Examples are algorithms to nd all rational, hyperexponential, geometric or Liouvillian solutions, to perform inde nite or de nite hypergeometric summation, to factorize linear operators, etc. (See for instance [5, 3, 7, 6].) Second, the algorithm that is described here has lower complexity than the usual algorithms, that are often based on undetermined coe cients. The approach here is to nd a degree bound on the solutions to be computed, and next nd recurrences to compute the coe cients of the solutions e ciently. The problem with undetermined coe cients arises with very concise equations having high degree solutions. Although the number of coe cients to be determined is high, the recurrences that are found by the new algorithm in [4] are of small order. The idea is to view the space K[x] as a subspace of a unusual space of formal power series, and to embed the space of polynomial solutions into a space of formal power series solutions.