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Formal Series and Linear Di erence Equations S. A. Abramov

Summary: Formal Series and Linear Di erence Equations
S. A. Abramov
Computing Centre
of Russian Academy of Sciences
Vavilova 40, 119991, Moscow GSP-1, Russia
A doubly in nite complex number sequence
(ck ) k 2 Z (1)
will be called a sequential solution of an equation of the form
ad(z)y(z + d)+ + a1(z)y(z + 1)+ a0(z)y(z) = 0 (2)
a1(z) a2(z) ::: ad;1(z) 2 C z], a0(z) ad(z) 2 C z] n f0g, if
ad (k)ck+d + + a1(k)ck+1 + a0(k)ck = 0
for all k 2 Z. A sequential solution (1) will be called subanalytic sequential
(or just subanalytic, for short), if the equation (2) has a solution in the form
of a single-valued analytic function f(z) such that ck = f(k) for all k 2 Z.
We discuss a way to compute the values of elements of a subanalytic
solution of equation (2) at arbitrary integer points, in particular at the
points where the polynomial ad (z ; d)a0(z) vanishes.
We show that the dimension of the C -linear space of all sequential so-
lutions of (2) is d, and for any integer m d there exists an equation


Source: Abramov, Sergei A. - Dorodnicyn Computing Centre of the Russian Academy of Sciences


Collections: Mathematics; Computer Technologies and Information Sciences