Solving Linear Differential Equations with Polynomial Coefficients in Some Power Series Classes

Sergei A. Abramov^{1} Computer Center of the Russian Academy of Science Vavilova 40, Moscow 117967, Russia abramov@ccas.ru, sabramov@cs.msu.su

Let a linear homogeneous ordinary differential equation Ly = 0 with polynomial coefficients be given. First we demonstrate how to find all points a (ordinary or singular of the given equation) and all formal power series

y_{a}(x) = ∑_{n=0}^{∞} a_{n}(x − a)^{n}

which satisfy Ly_{a}(x) = 0 and whose coefficients a_{n} have a “nice” explicit representation in terms of n. We consider the cases where a_{n} is a polynomial or a rational function of n, and where a_{n} is a hypergeometric term (i.e., a_{n+1}/a_{n} is a rational function of n). Hypergeometric series are in the latter class of power series.

Then we introduce the notion of the m-sparse power series (e.g., expanding sin x and cos x at x = 0 gives 2-sparse power series: a coefficient a_{n} of the series can be nonzero only if remainder(n, 2) is equal to a fixed number). We consider the problem of finding all m-points of a linear ordinary differential equation Ly = 0 with polynomial coefficients (i.e., the points at which the equation has a solution in the form of an m-sparse series). It is easy to find an upper bound for m. We prove that if m is fixed then either there exists a finite number of m-points and all of them can be found or all points are m-points and L can be factored L = L_{1} ∘ C where C is an operator of a special kind with constant coefficients. Additionally we formulate simple necessary and sufficient conditions for the existence of m-points for an irreducible L.

Combining the algorithms described in the paper with Petkovšek-Salvy’s algorithm we find all m-sparse m-hypergeometric solutions of the equation Ly = 0.