A New Approach for Computing Regular Solutions of Linear Difference Systems

In this paper, we provide a new approach for computing regular solutions of first-order linear difference systems. We use the setting of factorial series known to be very well suited for dealing with difference equations and we introduce a sequence of functions which play the same role as the powers of the logarithm in the differential case. This allows us to adapt the approach of [5] where we have developed an algorithm for computing regular solutions of linear differential systems. Introduction Let z be a complex variable and ∆ the difference operator whose action on a function f is defined by ∆(f(z)) = (z − 1) (f(z)− f(z − 1)). In the present paper, we consider first-order systems of linear difference equations of the form D(z)∆(y(z)) +A(z) y(z) = 0, (1) where D(z) and A(z) are given n×n matrices with factorial series entries of the form ∑ i≥0 ai z −[i] with ai ∈ C and z−[0] = 1, ∀ i ≥ 1, z−[i] = 1 z(z + 1) · · · (z + i− 1) , and y(z) is an n-dimensional vector of unknown functions of the complex variable z. We further assume that the matrix D(z) is invertible so that System (1) can be written as ∆(y(z)) = z B(z) y(z), (2) where q ∈ Z and B(z) is an n × n matrix whose entries are factorial series and B(∞) 6= 0. When q < 0, System (2) has a fundamental matrix of factorial series solutions. In the particular case q = 0, System (2) is said to be of the first kind