---
title: D’Arcais Polynomials and Their Applications
url: https://www.emergentmind.com/topics/d-arcais-polynomials-424a61a5-4181-46a8-a4fa-bc6ec106d5fa
type: topic
---

# D’Arcais Polynomials and Their Applications

D’Arcais polynomials—also called Nekrasov–Okounkov polynomials—form a central family of degree-$n$ polynomials $P_n(z)$ in one variable $z$, arising as the $q^n$-coefficients in the $-z$th power of Euler’s infinite product, $\prod_{m=1}^\infty (1-q^m)^{-z}$. These polynomials connect a wide array of mathematical subfields: they encode the combinatorics of partition hook-lengths, dictate the coefficients of Dedekind $\eta$-function powers, appear in modular forms and number-theoretic contexts, and give rise to rich algebraic and analytic phenomena via their roots, recursions, and specializations.

## 1. Definition, Recurrences, and Generating Functions

The D’Arcais polynomials are defined by the generating function
\[
\sum_{n=0}^\infty P_n(z)q^n = \prod_{m=1}^\infty(1-q^m)^{-z}
\]
for $|q|<1$ and $z\in\mathbb{C}$ [2010.07890, 2601.07103]. 

Equivalently,
\[
\sum_{n=0}^\infty P_n(z)\,q^n = \exp\left(z \sum_{m=1}^\infty \sigma_1(m)\frac{q^m}{m}\right),
\]
where $\sigma_1(m) = \sum_{d\mid m} d$ is the divisor sum function [2304.02694].

The polynomials satisfy a hereditary-type recurrence:
\[
P_n(z) = \frac{z}{n}\sum_{k=1}^{n} \sigma_1(k)\,P_{n-k}(z),\quad P_0(z)=1.
\]
This recurrence is linear with nonconstant coefficients, in contrast to classical convolution recursions. The combinatorial interpretation is encoded by the following hook-length (partition sum) formula:
\[
P_n(z) = \sum_{\lambda\vdash n} \prod_{h\in \mathrm{Hook}(\lambda)}\left(1+\frac{z}{h^2}\right),
\]
where the sum runs over integer partitions $\lambda$ of $n$, and $\mathrm{Hook}(\lambda)$ denotes the hook-length multiset of $\lambda$ [2010.07890, 1810.02226]. The shifted form gives a direct connection to the Nekrasov–Okounkov hook-length formula and arises in the context of Seiberg–Witten theory and random partitions.

For more generality, one considers the two-variable polynomials $P_n^{g,h}(x)$ attached to normalized arithmetic functions $g(\cdot)$, $h(\cdot)$ via
\[
P_n^{g,h}(x) = \frac{x}{h(n)}\sum_{k=1}^{n} g(k)\,P_{n-k}^{g,h}(x),\quad P_0^{g,h}(x)=1,
\]
with the D’Arcais case realized by $g(n)=\sigma_1(n)$ and $h(n)=n$ [2010.07890, 2010.05231].

## 2. Coefficients, Partition Sums, and Special Values

The coefficients $A(n,m)$ of $P_n(z) = \sum_{m=0}^{n} A(n,m)z^m$ admit explicit partition-sum descriptions:
\[
A(n,m) = \sum_{\lambda\vdash(n-m)}\prod_{i=1}^{\ell(\lambda)}\sigma_1(\lambda_i+1) \cdot \frac{\prod_{k=0}^{|\lambda|+\ell(\lambda)-1}(n-k)}{\prod_{i=1}^{\ell(\lambda)}(\lambda_i+1)},
\]
where $\lambda$ ranges over partitions of $n-m$, $\ell(\lambda)$ is the length, and $|\lambda|$ the sum of the parts [2010.07890, 2010.05231].

Key specializations include:
- $P_n(1) = p(n)$, the partition function.
- $P_n(-1)$ (up to sign), the number of partitions into distinct parts.
- $P_n(-24) = \tau(n)$, Ramanujan’s tau function [2004.06923].
- $P_n(k) = p_k(n)$, the number of $k$-colored partitions [2601.11226].

These evaluations link the polynomials to classical modular forms, with $\eta(\tau)^{-z}$'s Fourier coefficients identified as $P_n(z)$ (up to a $q$-exponential shift) [2010.07890]. Further, $E_4(\tau)^{-1}$ and $E_6(\tau)^{-1}$ reciprocals give $P_n(-240)$ and $P_n(504)$, respectively.

## 3. Root and Zero-Locus Properties

The location and nature of the zeros of D’Arcais polynomials have been central to several conjectures and results:
- Original conjectures due to Amdeberhan posited simplicity and real-negativity of roots, with reality first failing at $n=10$, where a pair of complex-conjugate roots appears [1810.02226].
- The revised conjecture, broadly supported numerically, posits that all nontrivial roots lie in the open left half-plane, i.e., every root $\neq0$ satisfies $\mathrm{Re}(z)<0$; such polynomials are Hurwitz-stable [1810.02226].
- The smallest real root of the auxiliary Volterra sequence $Q_n^g$ and its scaling relationship to the real zeros of $P_n^{g,h}$ gives fine control of zero locations. Transfer lemmas relate zeros via $y \approx x H(n-1)$, asymptotically capturing the lower and upper tail behavior [2304.02694].
- For the Chebyshev/Laguerre model, new explicit enclosing intervals for zeros of $L_m^{(1)}$ and $H_n$ have been obtained. For physicists’ Hermite polynomials $H_n(x)$, the real zeros satisfy $|x| \le \cos(\pi/(n+1))\sqrt{2n-2}$ [2304.02694].
- Roots of $P_n^\sigma(z)$ are excluded from broad families of cosets and algebraic numbers by algebraic-number-theoretic methods (Dedekind–Kummer theorem). For every $m\ge3$, all $n\ge1$, $P_n^\sigma(\zeta_m)\neq0$ for $\zeta_m$ any primitive $m$th root of unity, with further exclusions in cyclotomic and quadratic fields [2511.16276, 2509.06123].

## 4. Log-concavity, Unimodality, and Array Properties

D’Arcais polynomials and their coefficients display strong log-concavity and unimodality properties:
- The sequence of coefficients in $P_n(z)$ is ultra-log-concave for all $1\leq n\leq1000$ (checked explicitly), which in turn forces unimodality [1810.02226].
- Horizontal log-concavity (fixed $n$, varying $m$) is supported up to $n=1500$; vertical log-concavity (fixed $m$, varying $n$) generally fails globally but a weaker vertical $C$-log-concavity (up to $n\leq C m$) is expected [2010.05231]. 
- These log-concavity results are connected to Newton’s inequalities and the root distribution, and undergird monotonicity results such as strict decrease of $\sqrt[n]{p(n)}$ for $n\geq6$ (Sun’s conjecture and generalizations) [2601.11226].

## 5. Applications in Partition Theory, Modular Forms, and Asymptotics

D’Arcais polynomials underpin a range of arithmetic and combinatorial results:
- Modular lacunarity: Serre used values $P_n(r)$ to determine exactly which even $r$ yield lacunary powers $\eta(\tau)^{-r}$ [2010.07890].
- Lehmer’s conjecture, which states all non-trivial Fourier coefficients of $\eta(\tau)^{24}$ are non-zero, is equivalent to $P_n(-24)\neq0$ for all $n$ [2511.16276, 2509.06123].
- Sun’s conjecture on partition root monotonicity is reframed as a property of the largest real zero of certain D’Arcais polynomials, extending to $k$-colored, plane, and overpartitions [2601.11226].
- Connections to orthogonal polynomials: for prescribed arithmetic functions $g$ and $h$, the family $P_n^{g,h}$ includes as special cases the Chebyshev, associated Laguerre, Pochhammer, and, via partitions, MacMahon’s plane partition polynomials [2010.07890, 2304.02694].
- Probabilistic large deviations: The coefficients $A(2,n,k)$, when normalized, satisfy Bahadur–Rao type large deviation asymptotics for $k/n\to\kappa\in[0,1)$, governed by a Legendre–Fenchel transform determined by the abundancy index sequence $a_n=\sigma(n)/n$. This forms a bridge between the arithmetic of the divisor function and the probabilistic structure of weighted set partitions and commuting permutations [2601.07103].

## 6. Algebraic and Analytic Techniques for Non-vanishing and Zero-Detection

The zero-detection and non-vanishing theory for D’Arcais polynomials leverages deep algebraic number theory:
- Recurrences for the normalized polynomials $A_n^g(X)$ modulo small primes, especially $p=2,3$, show total splitting into linear factors, a key to proving non-vanishing on entire cosets in cyclotomic rings [2509.06123].
- Dedekind–Kummer theorem is used to connect the minimal polynomial of an algebraic integer $\alpha$ to the possible vanishing of $P_n^g(\alpha)$. Inert or ramified primes rule out vanishing by ensuring irreducibility mod $p$ [2511.16276].
- For the D’Arcais case ($g=\sigma$), explicit arithmetic congruence conditions exclude vanishing at roots of unity and their translates, as well as most elements in Gaussian and quadratic integer rings—leaving potential roots extremely sparse [2511.16276, 2509.06123].

## 7. Open Problems and Future Directions

Current avenues of research focus on:
- Establishing full horizontal log-concavity for all D’Arcais polynomials [2010.05231].
- Determining the optimal constant $C$ for vertical $C$-log-concavity (Hong–Zhang) [2010.05231].
- Complete characterization of root locations, especially the Hurwitz stability conjecture for $H_n^\sigma(X)=P_n^\sigma(X)/X$ [2511.16276].
- Extending large-deviation and partition asymptotic analysis to related polynomial families and interpreting zero-locus phenomena for general $(g,h)$ frameworks [2601.07103, 2601.11226].
- Advancing toward a proof of the Lehmer conjecture for all $n$ ($P_n^\sigma(-24)\ne0$) by further refining arithmetic exclusion arguments [2511.16276, 2509.06123].

D’Arcais polynomials thus constitute a nexus of partition combinatorics, modular form theory, algebraic number theory, and enumerative probabilistic asymptotics, with their properties continuing to fuel major developments across these disciplines.

Source: https://www.emergentmind.com/topics/d-arcais-polynomials-424a61a5-4181-46a8-a4fa-bc6ec106d5fa