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Graham, Knuth, and Patashnik (GKP), in their book Concrete Mathematics (1994) posed the following
"research problem" [Problem 6.94, pp. 319 and 564]:
Problem: Develop a general theory of the solutions to the recurrence
\[T(n,k) = (\alpha n + \beta k + \gamma) \, T(n-1,k) + (\alpha' n + \beta' k + \gamma') \, T(n-1,k-1)\]
for \(n\geq 1\) and \(k\in \mathbb{Z}\) with initial condition \(T(n,k) = \delta_{k0}\).
See Barbero--Salas--Villaseñor (2014, 2015) and Salas--Sokal (2021).
Let \(A = \{T(n,k)\}_{n,k\geq 0}\) be a triangular array satisfying the GKP recurrence. Let \(P_{n}(x) = \sum_{k=0}^n T(n,k) x^k\) denote its row-generating polynomial. Furthermore, we set \(\bmu = (\alpha,\beta,\gamma,\alpha',\beta',\gamma')\).
Conjecture: The sequence of polynomials \( \left(P_n(x)\right)_{n\geq 0}\) is coefficientwise Hankel totally positive in the seven variables \(\alpha,\beta,\gamma,\alpha',\beta',\gamma',x\).
In place of the usual GKP recurrence, consider instead the following recurrence: \[ \widetilde{T}(n,k) = (\widetilde{\alpha} (n-1) + \widetilde{\beta} k + \widetilde{\gamma}) \, \widetilde{T}(n-1,k) + (\widetilde{\alpha}' (n-1) + \widetilde{\beta}' (k-1) + \widetilde{\gamma}') \, \widetilde{T}(n-1,k-1) \] and let \(\widetilde{P}_n(x) = \sum_{k=0}^n \widetilde{T}(n,k) x^k\).
Conjecture: The sequence of polynomials \( \left(\widetilde{P}_n(x)\right)_{n\geq 0}\) is coefficientwise Hankel totally positive in the seven variables \(\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma},\widetilde{\alpha}',\widetilde{\beta'},\widetilde{\gamma'},x\).