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Our purpose is to asymptotically represent solutions of linear difference equations (k) when k ~ + 00 and A(k) is "small" by transforming them into so-called L-diagonal form. Two properties are then responsible for the asymptotic equivalence of an L-diagonal form to a diagonal one: a dichotomy condition on the diagonal part, and a growth condition on the perturbation term. In this manner, we derive some known asymptotic results from a central point of view and also several new extensions and generalizations of them. Some examples are constructed which demonstrate the need for a dichotomy-type condition, which shows incidentally that results of M. A. Evgrafov are incorrect, since they omit such a condition.

Abstract. In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data. §0. Introduction and main resultsWe consider the following "characteristic" Cauchy problem for the complex heat equationwhere τ = t + iη and z = x + iy are complex variables and ϕ(z) is assumed to be analytic in a domain D, which we assume to contain a neighborhood of the origin without loss of generality.As easily seen, this Cauchy problem has the unique formal solution. This formal solution diverges for general Cauchy data ϕ(z), as is shown by the example ϕ(z) = (1 − z) −1 , where By Cauchy's integral formula, for any r > 0 such that {|z| ≤ r} ⊂ D, we have the following estimate for {u n (z)} ∞ n=0 by taking some positive constants A and B, max |z|≤r |u n (z)| ≤ AB n n!, for all n = 0, 1, 2, . . . . (0.3) Such a formal power series is called Gevrey of order one in the variable τ .In the one variable case of ordinary differential equations, after the fundamental work of J. Ecalle, great progress has been made in the theory of summability for Gevrey type power series by many authors. As an application, they proved that every formal solution of an analytic ordinary differential equation is multi-summable (cf. on an asymptotic interpretation of formal solutions. Our purpose in this paper is to improve the asymptotic results by discussing the summation of formal solutions and under which conditions this is possible or not possible. While some of the results could possibly be extended to more general differential equations, we choose to consider only the heat equation since in this simple case we obtain both necessary and sufficient conditions.It is natural to ask the following questions about the formal solution u(τ, z) constructed above:[1] under what conditions on ϕ(z) does the formal solution (0.2) converge?[2] if (0.2) diverges, is it an asymptotic expansion of actual solutions as τ → 0 in sectorial domains?[3] if asymptotic solutions exist, when are they unique and how can they be constructed?As regards question [1], an easy way to get a sufficient condition for (0.2) to converges is to assume the Cauchy data...