On Integer Solutions of a Class of Differential-Difference Equations

Problems dealt with in a number of theoretical computer science matters are reduced to studying the solutions of algebraic differential-difference equations. By now, only linear equations of this sort have been studied to a fairly good extent. In the general case, a search for and an analysis of their solutions still involve insurmountable difficulties. Therefore, in a number of studies, solutions belonging to a predetermined class of functions (e.g. integer ones) are only considered. In Russia and abroad, results describing (to some or other degree) solutions for certain classes of algebraic differential equations have been obtained in this problem area. However, no results are available for nonlinear algebraic differential-difference equations, even for quite narrow classes of functions, e.g., polynomials. The article describes possible solutions (which are integer functions of a finite order) for nonlinear algebraic differential-difference equations of a fairly general kind. It is shown that equations of a certain structure can have integer solutions only in the form of quasipolynomials. The proof is based on using the techniques of dealing with integer functions that has recently been developed by the authors.