On the Log-Concavity of the D'Arcais Polynomials for Normalised Functions

Technical Summary: On the Log-Concavity of the D'Arcais Polynomials for Normalised Functions

Problem Statement
The paper investigates the log-concavity properties of D'Arcais polynomials (also known as Nekrasov–Okounkov polynomials), denoted as $P_n^g(X)$. These polynomials arise as the coefficients in the Fourier expansion of the infinite product $\prod_{n=1}^\infty (1-q^n)^{-f_g(n)X}$, where $g$ is a normalised arithmetic function ($g(1)=1$) and $f_g$ is a rescaled convolution of $g$ with the Möbius function. Specifically, the paper addresses the log-concavity of the coefficient sequences $p_n^g(k)$ (the coefficient of $X^k$ in $P_n^g(X)$).

The central problem concerns three distinct modes of log-concavity:

1. Horizontal Log-Concavity: The sequence $(p_n^g(k))_{k}$ is log-concave at $k$ if $(p_n^g(k))^2 \geq p_n^g(k+1)p_n^g(k-1)$. This relates to Conjecture 1.1 (Heim–Neuhauser) and Conjecture 1.2 (Abdesselam), which posit log-concavity for all $k \geq 1$. However, these conjectures are known to fail in specific cases (e.g., $\ell=2, k=2$).
2. Vertical Log-Concavity: The sequence $(p_n^g(k))_{n}$ is log-concave at $n$ if $(p_n^g(k))^2 \geq p_{n+1}^g(k)p_{n-1}^g(k)$.
3. Skew Log-Concavity: A new variant introduced in this work, defined for $n \geq k+1$ by the inequality $(p_n^g(n-k))^2 \geq p_{n+1}^g(n+1-k)p_{n-1}^g(n-1-k)$. This examines the log-concavity of coefficients along the "skew" diagonal of the triangular arrangement of polynomial coefficients.

Methodology
The author employs a combination of combinatorial analysis, recurrence relations, and asymptotic polynomial analysis.

• Renormalisation: The analysis focuses on the renormalised polynomials $A_n^g(X) = n! P_n^g(X)$, which have integer coefficients and simplify the recurrence relations.
• Recurrence Relations: The paper utilizes the recurrence relation derived from the differential operator applied to the generating function:
  $$a_n^g(k) = \sum_{m=1}^{n-k+1} \frac{(n-1)!}{(n-m)!} g(m) a_{n-m}^g(k-1)$$
  where $a_n^g(k)$ are the coefficients of $A_n^g(X)$.
• Polynomial Representation of Coefficients: A key methodological step involves proving that for a fixed $k$, the coefficient $a_n^g(n-k)$ behaves as a polynomial in $n$ (denoted $c_k^g(n)$) under specific conditions on the arithmetic function $g$. This is established via Lemma 4.1 and Proposition 4.3, which utilize the structure of the recurrence and the vanishing of $g(m)$ for small $m$.
• Asymptotic Analysis: The log-concavity inequalities are reduced to comparing the leading coefficients of the polynomials $c_k^g(n)$. The paper invokes Lemma 4.4, which states that for any polynomial $f(X)$ of degree $d \geq 1$, the expression $f(X)^2 - f(X+1)f(X-1)$ is dominated by a positive term $2d a_d^2 X^{2d-2}$ for large $X$, implying eventual log-concavity.

Key Contributions and Results

1. Horizontal and Vertical Log-Concavity (Theorem 2.1):
   The paper proves that for a normalised $\mathbb{Z}$-valued arithmetic function $g$ with $g(2) \neq 0$, the sequences of coefficients are eventually log-concave in both horizontal and vertical directions. Specifically, for any fixed $k$, there exist thresholds $n_h(g, k)$ and $n_v(g, k)$ such that for all $n$ beyond these thresholds, the polynomials are horizontally log-concave at $n-k$ and vertically log-concave at $n-k$. This result reinforces previous findings by Abdesselam et al. and Starr, providing a broader class of functions for which these properties hold asymptotically.

2. Introduction of Skew Log-Concavity (Theorem 2.2):
   The paper introduces the concept of skew log-concavity and establishes its validity under specific conditions:

   • If $g$ is a normalised $\mathbb{N}$-valued function where $g(\kappa) \neq 0$ and $g(m) = 0$ for $2 \leq m \leq \kappa$, then the polynomials are skewly log-concave at $k$ for sufficiently large $n$.
   • In the specific case where $g = e$ (the identity for Dirichlet convolution, where $e(1)=1$ and $e(n)=0$ for $n>1$), the polynomials are skewly log-concave everywhere.

3. Explicit Coefficient Formulas:
   Section 5 provides explicit polynomial formulas for the coefficients $a_n^g(n)$, $a_n^g(n-1)$, and $a_n^g(n-2)$ for general $g$ and specific cases like the identity function $E$, power functions $I_\ell$, and sum-of-divisors functions $\sigma_\ell$.

Significance and Claims
The paper claims to provide a "reinforcement" of previous results regarding the log-concavity of D'Arcais polynomials by establishing that these properties hold asymptotically for a wide family of arithmetic functions, even where global conjectures fail.

The primary novelty lies in the introduction and analysis of skew log-concavity. The author justifies this new perspective by noting that while previous research focused on coefficients starting from the bottom of the polynomial (low powers of $X$), the skew perspective (examining coefficients near the top, i.e., $X^{n-k}$) yields "remarkable simplicity" in theorems, particularly for the case $g=e$. The work demonstrates that the coefficients $a_n^g(n-k)$ are polynomial in $n$, a structural insight that allows the application of asymptotic arguments to prove log-concavity for large $n$.

The paper remains modest regarding the scope of these results, acknowledging that explicit counterexamples exist for small $n$ or specific parameters (as shown by Starr and others) and that the derived thresholds $n_h$ and $n_v$ are existence results rather than always providing tight, explicit bounds for all cases. The work concludes by suggesting further investigation into the interdependence of horizontal, vertical, and skew log-concavity.