A class of weighted Delannoy numbers

TL;DR

This paper investigates a generalized class of weighted Delannoy numbers, analyzing their asymptotic behavior and P-recursivity using advanced combinatorial and computational methods.

Contribution

It introduces a generalization of weighted Delannoy numbers with new initial conditions and applies Pemantle and Wilson's methods to determine their asymptotics.

Findings

Asymptotic formulas for the diagonal sequence $f_{n,n}$

Conditions for P-recursivity of the sequence

Extension of methods to broader classes of recurrence relations

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 $m n>0 $, with $f_{m,n}=\alpha^m \beta^n$ if $n m=0$. 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 $n m=0$. 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 $A,B,\alpha,\beta,\gamma\geq 0$. In addition, we also study its P-recursivity with the help of symbolic computation tools.