Formal Fundamental Solutions of Irregular Singular Differential Equations Depending Upon Parameters

Abstract

Given a system of linear differential equations with a pole, say at z = 0, it is well known that the system has a formal fundamental solution which is the product of a formal power series in a root of z, a matrix power of z, and the exponential of a polynomial in a root of z ⁻¹. Suppose that the system depends analytically upon several parameters in a neighborhood of some point in the parameter space. Then the question arises whether there exists an analytic formal fundamental solution, i.e., a formal fundamental solution whose coefficients are analytic in the parameters in a possible smaller neighborhood of the given point. In 1985, Babbitt and Varadarajan treated this problem together with that of the deformation of nilpotent matrices over rings. They assume that the exponential parts of the formal fundamental solutions are well behaved in some precise sense. In the present paper I will provide constructive proofs of their theorems on formal fundamental solutions and in this way also improve them slightly. Furthermore I will give a condition for well behaved exponential parts – and hence for the existence of an analytic formal fundamental solution – which can be expressed solely in terms of the given equation.