Rational Solutions of Linear Difference and Differential Equations

The talk presents an algorithm due to S.~Abramov that computes rational solutions of an equation of the type \begin{equation}\label{eq1} a_d(n)\,u_{n+d}+a_{d-1}(n)\,u_{n+d-1}+\dots+a_1(n)\,u_{n+1}+a_0(n)\,u_n=b(n), \end{equation} where the coefficients are polynomials in~$n$. This algorithm computes the solutions without performing any factorization, but only gcd computations. Thus it is rather independent of the ground field and in particular, it performs well when the ground field is some algebraic extension of~$\Q$. An analogous algorithm also due to S.~Abramov solves the differential case.