An Algorithm Computing the Regular Formal Solutions of a System of Linear Differential Equations

where x is a complex variable and A(x) a square matrix of dimension n the entries of which are formal meromorphic power series. Write A = x(A0 +A1x+ · · ·) (A0 6= 0) for the series expansion of A, where the coefficients are matrices over a subfield K of the field of complex numbers. There exists a basis of n formal solutions of the form (see, e.g. Turritin, 1955; Wasow, 1967) yi(t) = etzi(t) (i = 1, . . . , n), (1.2) where ti = x for positive integers ri, qi ∈ t−1K̄[t−1], λi ∈ K̄ and zi ∈ K̄[[t]][log(t)]. Here, K̄ denotes the algebraic closure of K. These solutions form the columns of a formal fundamental matrix solution of (1.1) which can be written as U(t) = H(t)te (1.3) with t = x for a positive integer r, H ∈ MnK̄[[t]] is a formal matrix power series, Λ ∈MnK̄ is a constant matrix with eigenvalues λ1, . . . , λn and Q = diag(q1, . . . , qn). The structure of the formal solutions depends on the nature of the origin as a singular point of the system. If Q = 0, then r = 1 and the system is called regular singular, otherwise irregular singular. In this latter case, one has necessarily q > 0. If q = 0, the singularity (or the system resp.) is said to be of the first kind (Wasow, 1967) or simple (Hartman, 1964). This is a sufficient condition for a regular singularity, and there is a standard method for the construction of the solutions in this case. An algorithmic