Determinantal representations for general Eulerian-type polynomial recurrences

Determine, for every sequence of polynomials \(\{\mathcal{P}_n(x)\}_{n\geqslant0}\) satisfying the general Eulerian recurrence \(\mathcal{P}_n(x)=(\alpha(x)n+\gamma(x))\mathcal{P}_{n-1}(x)+\beta(x)(1-x)\frac{\mathrm d}{\mathrm dx}\mathcal{P}_{n-1}(x)\), whether there exist functions \(f_n(x_0,\ldots,x_{n-1})\) and \(g_n(x_0,\ldots,x_{n-1})\) such that \(\mathcal{P}_n(x)=f_n(\mathcal{P}_0(x),\ldots,\mathcal{P}_{n-1}(x))\) and \(\frac{\mathrm d}{\mathrm dx}\mathcal{P}_n(x)=g_n(\mathcal{P}_0(x),\ldots,\mathcal{P}_{n-1}(x))\).

Background

The paper introduces the general Eulerian recurrence Pn(x)=(α(x)n+γ(x))Pn−1(x)+β(x)(1−x)ddxPn−1(x)\mathcal{P}_n(x)=(\alpha(x)n+\gamma(x))\mathcal{P}_{n-1}(x)+\beta(x)(1-x)\frac{\mathrm d}{\mathrm dx}\mathcal{P}_{n-1}(x), which includes the classical Eulerian polynomials as a special case. It asks whether both each polynomial Pn\mathcal{P}_n and its derivative can be expressed as functions of the preceding polynomials in the sequence.

The subsequent weighted-Hessenberg determinant method provides a partial answer to this problem and yields determinantal expressions in several examples, including Eulerian and second-order Eulerian polynomials. The general existence question for arbitrary sequences satisfying the recurrence is not fully resolved in the supplied paper.

References

For any sequence of polynomials ${\mathcal{P}n(x)}{n\geqslant0}$ that satisfies~Eulerian-type recurrence, whether there exist two functions $f_n(x_0,x_1,x_2,\ldots,x_{n-1})$ and $g_n(x_0,x_1,x_2,\ldots,x_{n-1})$ such that $$\mathcal{P}n(x)=f_n\left(\mathcal{P}_0(x),\mathcal{P}_1(x),\mathcal{P}_2(x),\ldots,\mathcal{P}{n-1}(x)\right),$$ $$\dx \mathcal{P}n(x)=g_n\left(\mathcal{P}_0(x),\mathcal{P}_1(x),\mathcal{P}_2(x),\ldots,\mathcal{P}{n-1}(x)\right).$$

Eulerian-type recurrence:

$\mathcal{P}_n(x)=(\alpha(x)n+\gamma(x))\mathcal{P}_{n-1}(x)+\beta(x)(1-x)\dx \mathcal{P}_{n-1}(x), $

— A general method of deducing the determinantal expressions for polynomial and its derivative  (2608.18670 - Lou et al., 19 Aug 2026) in Section 1, Problem \ref{problem0}