A class of weighted Delannoy numbers

Abstract

The weighted Delannoy numbers are defined by the recurrence relation $f_{m,n}=\alpha f_{m-1,n}+\beta f_{m,n-1}+\gamma f_{m-1,n-1}$ if $mn>0$, with $f_{m,n}={\alpha }^m{\beta }^n$ if $nm=0$. In this work, we study a generalization of these numbers considering the same recurrence relation but with $f_{m,n}=A^m{B}^n$ if $nm=0$. More particularly, we focus on the diagonal sequence $f_{n,n}$. 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 $A,B,\alpha ,\beta ,\gamma \ge 0$. In addition, we also study its P-recursivity with the help of symbolic computation tools.