Summary: Partial closed­form solutions of linear functional systems
Consider linear functional systems of the form `Y = BY where B is a known
matrix of coefficients, Y an unknown vector of functions and ` an operator such
as differentiation or (q­)difference. Depending on the operator and the coeffi­
cient domain, there are several known algorithms for constructing the solutions
of such systems in various classes of functions, such as polynomial, rational, hy­
perexponential or Liouvillian functions. But those algorithms only find solutions
Y whose components are all in the specified class.
We address the following related problem: given a subset fY e1 ; : : : ; Y em g
of the entries of Y and an appropriate (i.e., closed under the action of skew­
polynomials in `) class of functions, find all solutions Y whose speficied entries
are in the given class (more precisely we are interested in computing those entries
only). For example, given a differential system Y 0 = BY , find all the rational
functions that are Y 1 and Y 2 ­coordinates of some solution Y .
We present an algorithm that produces either one of two possible results:
ffl a proof that if the specified entries are in the given class, then all the other
entries must be in that class too. Or,
ffl a new system involving the specified entries only (and some of their ``deriva­
tives''), and whose solution space is exactly the projection on those entries
of the solutions of the initial system.

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

Collections: Mathematics; Computer Technologies and Information Sciences