Summary: A note on computing the regular solutions of linear di#erential
Sergei A. Abramov, Denis E. Khmelnov
Russian Academy of Sciences,
Dorodnicyn Computing Centre,
Vavilova 40, 119991, Moscow GSP1,
We present an approach to find all regular solutions of a system of linear ordinary di#erential equations
using EG # algorithm [2, 3] as an auxiliary tool.
L = Q # (z)D # + · · · +Q 1 (z)D +Q 0 (z), (1)
where D = d/dz.
Assume that Q # (x), . . . , Q 0 (x) are polynomials in z over C. A regular solution of the equation Ly = 0,
or, the same of the operator L, at a fixed point z 0 # C, is a solution of the form
(z - z 0 ) # F (z) (2)
with F (z) # C((z - z 0 ))[log(z - z 0 )], where C((z - z 0 )) is the field of Laurent series (here we do not consider
convergence problems; all series are formal). The value # is the exponent of regular solution (2). W.l.g. we