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We consider the second-order differential equation e2y″ e (1+e2ψ(x, e))y with a small parameter e, where ψ is even with respect to e. It is well known that it has two formal solutions y±(x, e) e e±x/eh±(x, e), where h±(x, e) is a formal series in powers of e whose coefficients are functions of x. It has been shown l4r that one resp. both of these solutions are 1-summable in certain directions if ψ satisfies certain conditions, in particular, concerning its x-domain. In the present article we give necessary (and sufficient) conditions for 1-summability of one or both of the above formal solutions in terms of ψ. The method of proof involves a certain inverse problem, i.e., the construction of a differential equation of the above form exhibiting a prescribed Stokes phenomenon with respect to e.

Consider the systems (1) \ (sI-B){dv \over ds} = (\rho I-A)v and (2) \ {dx \over dt} = (B+t^{-1} A)x, where s and t are variables, ρ is a parameter, and A and B e diag(λ1,…,λn) are n by n matrices. (1) has only regular singular points, and (2) has an irregular singular point at t e ∞. Several kinds of special solutions having particular behavior near singular points were selected in previous papers. In the present paper, the author shows how (2) results from (1) in a process of confluence as ρ → ∞. It is analyzed how the special solutions of (1) converge to those of (2) in that process. As a consequence new proofs of earlier results about connection problems are obtained.

Solutions of the vector difference equationy(x+e)−y(x−e)=2ef(x,y(x)),x being a complex variable and e>0 a small parameter, are constructed that are analytic onx-domains Ω which are independent of e. As the first case, horizontally convex bounded domains are considered, i.e., domains having the property that for eachx, x′∈Ω with the same imaginary part, the interval [x, x′] is contained in Ω; also considered are unbounded domains such as sectors open on the left or on the right. Using these results, it is shown that the Hausdorff distance between separatrices of certain systems of difference equations is exponentially small with respect to e. As an application, the so-called ghost solutions of the discretized logistic equation are considered in detail and, in particular, the lengths of the levels are estimated. Other applications, e.g., to the standard mapping, are presented.

Given a system of linear differential equations near an irregular singularity of pole type, formal invariants are quantities that remain unchanged with respect to linear transformations of the system. While certain “natural” formal invariants can easily be observed in formal fundamental solution matrices, the algorithms for constructing them do not readily show how the invariants can be universally described as properties of the coefficient matrix of the system, and in particular of the individual constant matrices in the power-series expansion. Other invariants have been abstractly defined by mapping properties of the differential operator, but they are not immediately related to either the natural invariants or the coefficients. In this paper we show how certain invariants in the formal solution may be described and calculated through matrix-theoretic properties of the coefficients and at the same time show how they are related to ones for the differential operator.