Combinatorial Recurrences and Linear Difference Equations

In this work we introduce the triangular arrays of depth greater than 1 given by linear recurrences, that generalize some well-known recurrences that appear in enumerative combinatorics. In particular, we focussed on triangular arrays of depth 2, since they are closely related to the solution of linear three-term recurrences. We show through some simple examples how these triangular arrays appear as essential components in the expression of some classical orthogonal polynomials and combinatorial numbers.