Search for Polynomial Solutions of Linear Functional Systems by Means of Induced Recurrences

The problem of searching for polynomial solutions of linear functional (differential, difference, and q-difference) equations is considered. The problem is solved by means of the construction of an induced recurrent system in the coefficients of the expansion of the desired solution in a certain basis. In the paper, practical aspects of the construction of such induced recurrences are discussed, and a new algorithm for the construction of polynomial solutions based on them is suggested.