Difference equations, pseudodifferential operators and semiclassical asymptotics

We discuss one approach which possibly allows one to construct the asymptotics of polynomials determined by some recurrent formulas given by 3 (or $2m+1$) diagonal matrices. The method is based on the objects appearing in semiclassical analysis and on the Maslov canonical operator in particular. These objects are quite elementary in 1-D case and any special knowledge is not needed.