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

Abstract

We study the triangular array $T(n,k;\mu )$ defined by the Graham--Knuth--Patashnik recurrences $$T(n,k)=(\alpha n+\beta k+\gamma )T(n-1,k)+({\alpha }^{\prime }n+{\beta }^{\prime }k+{\gamma }^{\prime })T(n-1,k-1)$$ with initial condition $T(0,k)={\delta }_{k,0}$ and parameters $\mu =(\alpha ,\beta ,\gamma ,{\alpha }^{\prime },{\beta }^{\prime },{\gamma }^{\prime })$, which are considered to be indeterminates. We first prove that, for any fixed $n\ge 0$, the sequence $(T(n,k;\mu ){)}_{k\ge 0}$ is strongly log-concave with the coefficientwise partial order in the variables $\alpha ,\beta ,\gamma ,{\alpha }^{\prime },{\beta }^{\prime },{\gamma }^{\prime }$. Moreover, we show that the sequence of the corresponding row-generating polynomials $(P_n(x;\mu ){)}_{n\ge 0}$ is strongly log-convex with the coefficientwise partial order in the variables $x$ and $\alpha ,\beta ,\gamma ,{\alpha }^{\prime },{\beta }^{\prime },{\gamma }^{\prime }$. Finally, we show that this sequence is coefficientwise Hankel-totally positive of order 2 with the same partial order.

pdflatex, 16 pages. No figures