Concavity Properties of Numbers Satisfying the Binomial Recurrence

Abstract

We consider square arrays of numbers $\{a(n,k)\}$, generalizing the binomial coefficients:
$a(n,0)=c_n$, where the $c_n$ are non-negative real numbers; $a(0,k)=c_0$, and if $n,k>0$, then $a(n,k)=a(n,k–1)+a(n–1,k)$.
We give generating functions and arithmetical relations for these numbers. We show that every row of such an array is eventually log concave, and give a few sufficient conditions for columns to be eventually log concave. We also give a necessary condition for a column to be eventually log concave, and provide examples to show that there exist such arrays in which no column is eventually log concave.