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arXiv · 2607.04217

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

Jesús Salas
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Abstract

We study the triangular array $T(n,k;\mu)$ defined by the Graham--Knuth--Patashnik recurrences $$ T(n,k) \;=\; (\alpha n + \beta k + \gamma) \, T(n-1,k) + (\alpha' n + \beta' k + \gamma') \, T(n-1,k-1) $$ with initial condition $T(0,k)=\delta_{k,0}$ and parameters $\mu=(\alpha,\beta,\gamma,\alpha',\beta',\gamma')$, 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 $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. 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 $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. Finally, we show that this sequence is coefficientwise Hankel-totally positive of order 2 with the same partial order.

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Jesús Salas. 2026-07-05. Log-concavity and log-convexity in the theory of the Graham--Knuth--Patashnik recurrences. https://arxiv.org/abs/2607.04217

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