Minimal completely factorable annihilators
Sergei A. Abramov, Eugene V. Zima · 1997
We propose an algorithm to construct the minimal annihilating operator of a function or a sequence, when the operator is completely factorable (i.e. can be decomposed in first order factors).The algorithm is designed in the frame of the Ore rings theory and can be used in the differentiaf, difference and q-difference cases.We describe also a Maple implement ation of the algorithm.1 Introduction Constructing a linear ordinary differential operator annihilating a function (an annihilator of the function) is necessary when solving many computer algebra problems.We list some of these problems.P1. Expanding a function as a power series and subsequently investigating the expansion.An annihilator lets one construct the recurrence for the series coefficients and manipulate them ([14, 17]). P2.Solving linear inhomogeneous equations.Some methods use annihilators of the right-hand side ([4, 8]).P3. Integrating.If the minimal annihilator L, ord L = n, of j is given, then one can check whether there exists a primitive of ~with an n-th order minimal annihilator.If yes, then it is possible to express the primitive explicitly via f ([91) P4.Recognizing the equivalence of two given functions.If the common annihilator of both the functions is given, then it suffices to check the agreement between the corresponding "initial conditions" (a classical approach).The minimal annihilator, i.e. the annihilator of the lowest order, is the most informative.Note that to solve P3 only the minimal annihilator of j is suitable.Applying algorithm [8] to an equation with a d'Alembertian righthand side guarantees that all d'Alembertian solutions will be found only in the situation when the minimaf annihilator of the right hand side, decomposed in first order factors, is given.(A function is d'Alembertian if it has a completely -Work reported herein was supported in part by RFBR under Grant 95-01-01138.