An Efficient Algorithm for Deriving Summation Identities from Mutual Recurrences

This paper closes an algorithmic problem of summing a set of mutual recurrence relations with constant coefficients. Given an order d system of the form A(n) = ∑d i=1 MiA(n− i)+G(n), where A,G : N→ Km and M1, . . . ,Md ∈Mm(K) for some field K and natural number m, this algorithm computes the sum ∑n i=0 A(i) as a K-linear combination of A(n), . . . , A(n − d), the initial conditions and sums of the inhomogeneous term G(n). The runtime of this algorithm is shown to be polynomial in m and d.