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On Solutions of Linear Functional Systems Sergei A. Abramov Manuel Bronstein
 

Summary: On Solutions of Linear Functional Systems
(Errata)
Sergei A. Abramov Manuel Bronstein
The bounds given in the paper are valid when the system of recurrences (5)
and the equations (6) are valid for all n 2 Z, which is the case when using the
power basis P = hx n i n0 , because it can be extended to negative values of n.
Therefore the bounds given in the paper are valid for di erential and q-di erence
equations provided that the basis P is used to produce the recurrence.
In the case when the basis used is valid only for n   for some  2 Z
(for example we can have  = 0 for di erence equations), then the system of
recurrences (5) and the equations (6) are valid only for n  . Since we apply (6)
to n = N s in the proof of Theorem 4, that proof is valid only when N s  ,
i.e. N  s + . Therefore, the correct version of Theorem 4 is the following,
where deg(0) = 1 by convention:
Theorem 4 Let L be an r m matrix with entries in EndB (K[x]), F 2 K[x] r ,
Y 2 K[x] m be nonzero and N = max i fdeg Y i g. If LY = F then either
N  s + maxf 1; max i fdeg(F i )gg or Ker(M s (N s)) 6= 0, where M s is
as in (6) and  is either 1 or an integer such that the equations (6) are valid
only for n  .
When the basis P is used, then the transformed recurrences remain valid for

  

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

 

Collections: Mathematics; Computer Technologies and Information Sciences