Log-concavity and log-convexity in the theory of the Graham--Knuth--Patashnik recurrences

Abstract

We study the triangular array $T(n,k;μ)$ defined by the Graham--Knuth--Patashnik recurrences $$ T(n,k) \;=\; (αn + βk + γ) \, T(n-1,k) + (α' n + β' k + γ') \, T(n-1,k-1) $$ with initial condition $T(0,k)=δ_{k,0}$ and parameters $μ=(α,β,γ,α',β',γ')$, which are considered to be indeterminates. We first prove that, for any fixed $n\ge 0$, the sequence $(T(n,k;μ))_{k\ge 0}$ is strongly log-concave with the coefficientwise partial order in the variables $α,β,γ,α',β',γ'$. Moreover, we show that the sequence of the corresponding row-generating polynomials $(P_n(x;μ))_{n\ge 0}$ is strongly log-convex with the coefficientwise partial order in the variables $x$ and $α,β,γ,α',β',γ'$. Finally, we show that this sequence is coefficientwise Hankel-totally positive of order 2 with the same partial order.