Matrices in Combinatorial Problems

Abstract

Consider the difference equation (also called recurrence relation) with given boundary conditions

$$ {u_{n + k}} = {a_{}}{u_{n + k - 1}} + {a_2}{u_{n + k - 2}} + \ldots + {a_k}{u_n} + {b_n}$$

(4.1)

$${u_i} = {c_l},0 \leqslant l \leqslant k - 1$$

(4.2)

where the constants a ₁, ...a _k , c ₀ , ...,c _k−1 and the sequence 〈b _n 〉 are given. A solution to this equation is a sequence 〈u _n 〉 satisfying (4.1) and (4.2). If b _n = 0, for all n, then the resulting equation is the corresponding homogeneous equation to (4.1).