A general method of deducing the determinantal expressions for polynomial and its derivative

Abstract: In this paper, we present a general method of deducing the determinantal expressions for a polynomial and its derivative. As illustrations, we provide three determinantal expressions for the derivative of the Eulerian polynomial. Using a functional equation discovered by Gessel,we also establish the determinantal expressions for the second-order Eulerian polynomial and its derivative.

• The paper develops a general kernel-series framework that converts weighted generating functions of the form 1/(1−K) into lower Hessenberg determinants for polynomial sequences and their derivatives.
• The method produces new and unified determinant formulas for Eulerian, derangement, and second-order Eulerian polynomials, including derivative representations based on logarithmic generating-function derivatives.
• The paper connects these formulas to combinatorics by interpreting derivatives as marked descents or plateaux, while leaving open whether all sequences satisfying the general Eulerian recurrence admit such representations.

Overview

This paper develops a general framework for producing determinantal expressions for a polynomial sequence and its derivatives, and applies it to the Eulerian polynomials, derangement polynomials, and second-order Eulerian polynomials. The core objects are lower Hessenberg determinants whose entries are generated by a "kernel" series, and the derivative is handled by replacing the first column with coefficients of the logarithmic derivative of the generating function. The work is motivated by the general Eulerian recurrence of Hwang, Chern and Duh,

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

and by the question of whether $\mathcal{P}_n(x)$ and its derivative can always be written as functions of the earlier polynomials in the sequence (2608.18670). The framework gives a partial affirmative answer.

The weighted Hessenberg framework

The starting point is the classical expansion of Hessenberg determinants due to Cahill et al.: when the superdiagonal entries of a lower Hessenberg matrix $H_n$ are all $-1$, its determinant satisfies $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$. Given a weight sequence $(\rho_n)$ with $\rho_0=1$ and polynomials $K_m(x)$, the authors define a weighted Hessenberg matrix $H_n(x)$ whose $(n,r)$ entry is $\mathcal{P}_n(x)$0 for $\mathcal{P}_n(x)$1, with $\mathcal{P}_n(x)$2 on the superdiagonal.

The key structural result (Lemma 2.1) is a generating-function identity: if $\mathcal{P}_n(x)$3 and $\mathcal{P}_n(x)$4, then

$\mathcal{P}_n(x)$5

The proof is a one-line convolution: expanding along the last row and reindexing yields $\mathcal{P}_n(x)$6. Consequently, any polynomial sequence whose ordinary (weighted) generating function is of the form $\mathcal{P}_n(x)$7 admits a Hessenberg determinant representation with entries determined by the kernel $\mathcal{P}_n(x)$8. This recovers, by choosing $\mathcal{P}_n(x)$9 from the exponential generating function of the Eulerian polynomials, Chow's Hessenberg determinant for $H_n$0, with $H_n$1; and, using Brenti's generating function for derangement polynomials $H_n$2, a new determinant for $H_n$3 with $H_n$4.

Determinants for the derivative

The main theorem addresses derivatives. Define the logarithmic derivative

$H_n$5

Since $H_n$6, one has $H_n$7, i.e. $H_n$8 itself satisfies a convolution recursion and hence admits a Hessenberg determinant of its own. The main result (Theorem 2.1) states that

$H_n$9

where $-1$0 is obtained from $-1$1 by replacing the first column entries $-1$2 with $-1$3, while all other columns are unchanged. The proof compares the recursion satisfied by $-1$4 (via last-row expansion) with the coefficient recursion derived from $-1$5; both agree with the same initial condition. This is the "general method" of the title: once the kernel $-1$6 is known, both $-1$7 and $-1$8 are determinantal, and $-1$9 is itself computable as a determinant involving $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$0.

Applied to the Eulerian polynomials, where

$\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$1

the coefficient identity $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$2 yields two explicit determinants: one expressing $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$3 in terms of $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$4 on a fixed kernel $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$5, and a second expressing $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$6 purely from the kernel entries, whose last-row entries are $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$7, $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$8, etc. Notably, the second determinant involves no derivatives at all — the derivative is encoded entirely in the kernel structure.

A combinatorial determinant via insertion of descents

The paper also derives a third, structurally different determinant for $\det H_n=\sum_{r=1}^{n}h_{n,r}\det H_{r-1}$9. Combinatorially, $(\rho_n)$0 marks a chosen descent, and inserting $(\rho_n)$1 immediately after a descent of $(\rho_n)$2 gives a bijection with permutations in $(\rho_n)$3 in which $(\rho_n)$4 sits between two entries in decreasing order, so

$(\rho_n)$5

confirming Gessel's OEIS observation that $(\rho_n)$6 enumerates permutations of $(\rho_n)$7 starting with an ascent. A case analysis on the decomposition $(\rho_n)$8 (position of the letter 1, with $(\rho_n)$9 inserted into $\rho_0=1$0 or $\rho_0=1$1) yields the convolution identity

$\rho_0=1$2

Feeding this recursion into the Hessenberg expansion produces a determinant whose first column is $\rho_0=1$3 and whose remaining entries involve binomial coefficients times $\rho_0=1$4-values, with diagonal $\rho_0=1$5. The $\rho_0=1$6 instance evaluates to $\rho_0=1$7, matching the known coefficients.

Second-order Eulerian polynomials

The final section treats the second-order Eulerian polynomials $\rho_0=1$8, the plateau (equivalently, ascent or descent) enumerators over Stirling permutations of $\rho_0=1$9. The tool here is Gessel's functional equation for the exponential generating function $K_m(x)$0:

$K_m(x)$1

Writing $K_m(x)$2 and differentiating with respect to $K_m(x)$3 gives $K_m(x)$4, whence

$K_m(x)$5

So the kernel is $K_m(x)$6 and $K_m(x)$7 for $K_m(x)$8, and the general lemma immediately yields a lower Hessenberg determinant of order $K_m(x)$9 for $H_n(x)$0 whose first column is $H_n(x)$1 and whose remaining columns are binomial multiples of earlier $H_n(x)$2-values. The authors note this recovers, via a generating-function argument, a result recently obtained from a recursion by Ma, Liu, Yeh and Yeh — and the functional-equation proof is arguably more transparent than the recursion-based one.

Applying the derivative theorem with $H_n(x)$3 gives determinants for $H_n(x)$4, with first column given by $H_n(x)$5, itself a determinant whose first column is $H_n(x)$6; for instance $H_n(x)$7 evaluates to $H_n(x)$8. Combinatorially, inserting the two copies of $H_n(x)$9 right after a plateau of $(n,r)$0 shows

$(n,r)$1

where $(n,r)$2 consists of Stirling permutations of $(n,r)$3 in which the two copies of $(n,r)$4 are adjacent with equal neighbors on both sides — a plateau-analogue of the classical descent-insertion picture.

Scope and open questions

The framework applies whenever the (weighted) ordinary generating function of the sequence can be written as $(n,r)$5 with a tractable kernel; sequences governed by nonlinear functional equations, such as $(n,r)$6, require an additional integration step to extract the kernel, which is available here only because Gessel's equation is explicitly solvable in this form. The authors pose, but do not resolve, the full Problem 1.1: whether every sequence satisfying the general Eulerian recurrence admits such determinantal representations for both the polynomial and its derivative. The paper also does not address whether these determinants offer computational or structural advantages (e.g., for zero distribution or stability questions of the type studied by Haglund and Visontai), leaving that connection unexplored.

Conclusion

The paper contributes a compact and reusable mechanism — kernel series, Hessenberg determinants, and logarithmic differentiation — that simultaneously produces determinantal formulas for a polynomial and its derivative. It unifies existing Hessenberg representations of Eulerian and derangement polynomials, adds three distinct determinants for $(n,r)$7 including a purely derivative-free one, and extends the method to the second-order Eulerian polynomials via Gessel's functional equation, together with a combinatorial interpretation of $(n,r)$8 in terms of marked plateaux.