Summary: D'Alembertian Solutions
of Linear Di#erential and Di#erence Equations
Sergei A. Abramov
Computer Center of
the Russian Academy of Science,
Vavilova 40, Moscow 117967, Russia.
Department of Mathematics and Mechanics,
University of Ljubljana,
Jadranska 19, 61111 Ljubljana, Slovenia.
D'Alembertian solutions are those expressible as nested indefinite integrals resp.
sums of hyperexponential functions. They are a subclass of Liouvillian solutions, and
can be constructed by recursively finding hyperexponential solutions and reducing the
order. Knowing d'Alembertian solutions of Ly = 0, one can write down the corre
sponding solutions of Ly = f and L # y = 0.
Let k be a field of characteristic zero, X an indeterminate over k, # a nonzero field