Analytic theory of singular difference equations

Abstract

Definition 6. An operator L, (y) (or equation Ln (y)--o) which is proper in G will be called completely proper in G if, in G, proper fundamental sets of solutions exist which are connected by proper periodic functions.Definition 7. A set Q, (x) , . . .q,, (x) has a point of division in G if this set can be separated into two groupsAnalytic Theory of Singular Difference Equations.q, (x),... Q,,(x); Q~+~ (~),... Qn(x)Definition 8. Let G denote a subregion of F, the lower boundaries of G and F being coincident.An operator L,,(y) (or equation L~(y)= o) with coefficie~ts of the same kind, in G, as in (I) and (2) will be said to bebelonging to L~ (y), has a point of divi,~ion in G.Definition 9. Let F be a curve extending to infinity and lying in I. Let RF be the portion of F between F and the right boundary of F. The curve 9 and the region RF will be termed proper for the set Q, (x), Q, (~) if this set is proper along F and ah'o in RF.2. Let us writeSome Properties of B' and Proper Curves.With a system (or equation) of order n we have associated n polynomials, 1_ Pj(x) (j--~ I, ... n), in xP, of degree not greater than p and without constant terms.These polynomials occur in the exponential parts of the formal series (7, (7 a); w I).In this connection subscripts I to n are attached to p, m, r, 7, (I, ... v so as to differentiate between the n exponential factors which accompany the n formal power series (when we consider a single equation)or so as to differentiate between the n columns of the formal matrix (when a system is considered).Unless stated otherwise, we shall take for each of these elements a common value of p and m; this can always be effected by taking these integers