Enumeration of lattice paths using Vieta polynomials

Dyck-type lattice paths, consisting of up and down diagonal steps on the integer lattice, can be either unconfined ? if the only restriction is to remain in the non-negative half-plane ? or confined, if, in addition, they are not allowed to pass above a fixed horizontal line. In this paper, we establish a formula for the number of unconfined Dyck-type lattice paths of a given length, and also investigate the enumeration of confined Dyck-type paths, deriving recurrence relations involving Vieta-Fibonacci polynomials.