Root location for the characteristic polynomial of a Fibonacci type sequence

We analyse the roots of the polynomial xn − pxn−1 − qx − 1 for p≽ q ≽ 1. This is the characteristic polynomial of the recurrence relation Fk,p,q(n) = pFk,p,q(n − 1) + qFk,p,q(n − k + 1) + Fk,p,q(n − k) for n ≽ k, which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.