Linear vector recursions of arbitrary order

Solution of various combinatorial problems often requires vector recurrences of higher order (i.e., the order is larger than 1). Assume that there are given matrices A 1 , A 2 , . . . , A s , all from C k × k . These matrices allow us to define the vector recurrence ¯ v n = A 1 ¯ v n − 1 + A 2 ¯ v n − 2 + · · · + A s ¯ v n − s for the vectors ¯ v n ∈ C k , n ≥ s . The paramount result of this paper is that we could separate the component sequences of the vectors and find a common linear recurrence relation to describe them. The principal advantage of our approach is a uniform treatment and the possibility of automatism. We could apply the main result to answer a problem that arose concerning the rows of the modified hyperbolic Pascal triangle with parameters { 4 , 5 } . We also verified two other statements from the literature in order to illustrate the power of the method.