A class of weighted Delannoy numbers

José María Grau, Antonio M Oller-Marcen, Juan Luis Varona


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 $n m=0$. In particular, we focus on the diagonal sequence $f_{n,n}$ for which we find its asymptotic behavior and we study its P-recursivity.


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