Recurrence relations, succession rules and the positivity problem

In this paper we present a method to pass from a given recurrence relation with constant coefficients (in short, a C-finite recurrence) to a finite succession rule defining the same number sequence. Our method consists in two steps: first, we transform the given recurrence relation into an extended succession rule, then we provide a series of operations to reduce such an extended succession rule into an ordinary succession rule, equivalent to the previous one. As a byproduct, our method can be used to investigate the positivity of a C-finite recurrence.