A class of weighted Delannoy numbers

TL;DR

This work generalizes weighted Delannoy numbers by allowing axis boundary conditions $f_{m,0}=A^m$ and $f_{0,n}=B^n$ while maintaining the same interior recurrence. It provides explicit generating functions for both the bivariate and the diagonal sequences, and leverages PW-type asymptotic methods (with Melczer) to derive central-diagonal growth rates, including phase-transition behavior dependent on parameter ratios. The diagonal sequence is proven to be $P$-recursive (holonomic), with a 4-term recurrence in general (and simpler forms in special cases), and the analysis is complemented by symbolic-computation support. The paper also outlines several natural extensions, including negative parameters and alternative initial conditions, highlighting rich future directions for enumeration and asymptotics of generalized Delannoy-type numbers.

Abstract

The weighted Delannoy numbers are defined by the recurrence relation $f_{m,n}=αf_{m-1,n}+ βf_{m,n-1}+ γf_{m-1,n-1}$ if $m n>0 $, with $f_{m,n}=α^m β^n$ 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 $A,B,α,β,γ\geq 0$. In addition, we also study its P-recursivity with the help of symbolic computation tools.

Figures (5)

• Figure 3: For $A=B=1$, $\alpha=2$, $\beta=1$ and $\gamma=3$ in \ref{['eq:recu']}, $f_{2,2}=56$
• Figure 4: $F_n\rightarrow 21.14\dots$ for $\{A,B,\alpha,\beta,\gamma\} = \{2, -4, -4, 3, 21\}$
• Figure 5: $F_n$ is unbounded for $\{A,B,\alpha,\beta,\gamma\} = \{2, -4, -4, 3, 431/20\}$
• Figure 6: $F_n$ tends to the $2$-cycle $\{1.81\dots,-2.20\dots\}$ for $\{A,B,\alpha,\beta,\gamma\} = \{27/20, -27/20, 1, 1, -2\}$
• Figure 7: $F_n$ tends to the $3$-cycle $\{2.83\dots,0.59\dots,-4.76\dots\}$ for $\{A,B,\alpha,\beta,\gamma\} = \left\{8/5, -8/5, 1, 3/2, -2\right\}$

Theorems & Definitions (63)

• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 1
• Corollary 1
• Lemma 1
• proof
• Proposition 2
• proof