Laplace integrals and factorial series in the theory of linear differential and linear difference equations

Abstract

and do(x), dn(x) 0O. In view of the fact that the two mentioned fields are to a considerable degree analogous, it has been found convenient to give the developments for both of these in a single paper. Another outstanding method, a method which previously had been applied with complete success in the two indicated fields, as well as in the field of q-difference equations, is that based on the study of asymptotic properties of solutions. In this connection the starting point is given by the full sets of formal series solutions (in general divergent), which are known to exist in all cases. Existence of a complete set of formal series solutions for a difference equation (B), as well as other results of a formal character, have been established by G. D. Birkhoff.t For differential equations (A) existence of a full set of formal solutions follows from a work of E. Fabry.1 In the