 
Summary: Indicial Rational Functions of Linear Ordinary
Dierential Equations with Polynomial Coecients
S. A. Abramov, A. A. Ryabenko
gomputing gentre of the ussin edemy of ienes
wosowD IIWWWID qEID vilovD RH
sabramov@ccas.ruD ryabenko@cs.msu.ru
Abstract
The notion of indicial rational function is introduced for ordinary dierential equations
with polynomial coecients and polynomial righthand sides, and the algorithms for its
construction are proposed.
1 Introduction
st is known tht if n nlyti solution of di'erentil eqution
ad(x)y(d)
(x) + · · · + a1(x)y (x) + a0(x)y(x) = 0, @IA
where a0(x), a1(x), . . . , ad(x) re polynomils over CD hs singulrity @in prtiulrD poleA in
point D then ad() = 0 @we ssume tht ad(x) is nonEzero polynomilAF yne n ompute
lower ound of the order of the pole using the lest integer root of the indiil equtionF his
eqution is n lgeri eqution of degree not exeeding dD nd it orresponds to eqution @IA
nd to the point SD TF sf the indiil eqution hs no integer rootD then eqution @IA hs
no nonEzero solutions tht either re regulrD or hve pole t F essume tht every indiil
