Summing Symbols in Mutual Recurrences

Abstract. The problem of summing a set of mutual recurrence relations with constant coefficients is investigated. A method is presented for summing an order d system of the form for some field K and natural number m. The procedure expresses the sum n i=0 A(i) in terms of A(n), . . . , A(n − d), initial conditions and sums of the inhomogeneous term G(n). Problem Statement An important task in computer algebra systems is evaluating indefinite sums, that is computing values s n = n k=0 a k where the a k are some sequence depending only on k. Today many functions can be summed, in part due to the pioneering work of many researchers [3], [5], One area of interest is summing recurrence relations. Summing any a k is a special case of computing the value of A n where A n = A n−1 + a k and A 0 = 0. Recurrence relations arise frequently in algorithm analysis and numerical analysis of differential equations. The classical example is the Fibonacci sequence, defined as a function f : N → N 3 given by F (n) = F (n − 1) + F (n − 2) ∀n ≥ 2 with F (0) = 0, F (1) = 1. It is well known 4 that this sequence satisfies the property n i=0 F i = F n+2 − 1. This identity is nice because it presents the sum in terms of the original Fibonacci symbol. An even trickier situation is a system of linear recurrences, often referred to as mutual recurrences in the literature. Consider the following example: A, B : N → Q satisfy A(n + 2) − A(n + 1) − A(n) − B(n) = 1 and −A(n)+

Summing symbols in mutual recurrences

The problem of summing a set of mutual recurrence relations with constant coefficients is investigated. A method is presented for summing an order d system of the form A(n) = ∑i=1d MiA(n-i) + G(n), where A,G:ℕ → Km and M1...,Md ∈ Mm (K) for some field K and natural number m. The procedure expresses the sum ∑i=0n in terms of A(n),...,A(n-d), initial conditions and sums of the inhomogeneous term G(n). © 2011 Springer-Verlag