Proof of two conjectures of Petkovšek and Wilf on Gessel walks

Let F ( m ; n 1 , n 2 ) be the number of Gessel walks with exactly m steps ending at the point ( n 1 , n 2 ) . In this paper a probabilistic model of Gessel walks is established and F ( m ; n 1 , n 2 ) is shown to be the number of pairs of non-crossing Dyck paths and free Dyck paths. Two formulas for F ( 2 n + 2 k ; 0 , n ) and F ( n + 2 k ; n , 0 ) conjectured by Petkovšek and Wilf are proved.