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.