Integrability test for evolutionary lattice equations of higher order

A generalized summation by parts algorithm is presented for solving of difference equations of the form Tm(y)−ay=bTm(y)−ay=b with variable coefficients, where T   denotes the shift operator. Solvability of equations of this type with respect to the coefficients of formal symmetry (or formal recursion operator) provides a convenient integrability test for evolutionary differential-difference equations u,t=f(u−m,…,um)u,t=f(u−m,…,um).