Analytic theory of linear q-difference equations

Abstract

Introduction.Setting of the problem.Formal q-summation.Lemmas on analytic q-summation.Factorization.The Preliminary Lemma.The Fundamental Existence Theorem.Connection between the 'left' and the 'right' solutions. i. Introduction.The subject of this paper is to develop the analytic theory of a q-difference systemor, which is an essentially equivalent matter, the analytic theory, of a single q-difference equation ( I a) L.(Y) =---Y(q TM) + al (x)'y(q n-l~) +"" + an(x) y(x) = o (~n(X) ~ o).It as assumed that the coefficients a(x)in (I) or (I a) are analytic for Ix[ =~ Q, being representable for these values of x as follows