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\centerline{\bf Solving Linear Differential Equations with Polynomial}
\centerline{\bf Coefficients in Some Power Series Classes}

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\centerline{Sergei A.~Abramov\thanks{Supported in part by
the RFBR and INTAS under grant 95-IN-RU-412.}}
\centerline{Computer Center of the Russian Academy of Science}
\centerline{Vavilova 40, Moscow 117967, Russia}
\centerline{abramov@ccas.ru, sabramov@cs.msu.su}

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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
$$y_a(x) = \sum _{n=0}^\infty a_n (x-a)^n$$
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 $L=L_1 \circ C$ 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
Petkov\v sek-Salvy's algorithm we find all $m$-sparse
$m$-hypergeometric solutions of the equation $Ly=0$.

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