A class of weighted Delannoy numbers

The weighted Delannoy numbers are defined by the recurrence relation fm,n = α fm−1,n +β fm,n−1 + γ fm−1,n−1 if mn > 0, with fm,n = α mβ n if nm = 0. In this work, we study a generalization of these numbers considering the same recurrence relation but with fm,n = A mB n if nm = 0. More particularly, we focus on the diagonal sequence fn,n. With some ingenuity, we are able to make use of well-established methods by Pemantle and Wilson, and by Melczer in order to determine its asymptotic behavior in the case A, B, α, β, γ ≥ 0. In addition, we also study its P-recursivity with the help of symbolic computation tools