Partial closedform solutions of linear functional systems Consider linear functional systems of the form `Y = BY where B is a known 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.