---
title: D’Arcais Polynomials & Modular Forms
url: https://www.emergentmind.com/topics/d-arcais-polynomials
type: topic
---

# D’Arcais Polynomials & Modular Forms

D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, are a central family of recursively defined polynomials intimately connected to the partition theory, the arithmetic and combinatorial properties of the Dedekind eta function, and the structure of modular forms. Originally introduced by Francesco D’Arcais (1913), they encode deep arithmetic information, relate to multiple classical number-theoretic sequences, and have found broad applications ranging from statistical mechanics to random partition theory and asymptotic combinatorics.

## 1. Definitions and Fundamental Properties

D’Arcais polynomials \( P_n(z) \) are classically defined via the generating function:
\[
\prod_{m=1}^\infty (1 - q^m)^{-z} = \sum_{n=0}^\infty P_n(z)\,q^n.
\]
Here, \( q = e^{2\pi i \tau} \) and \( \tau \) lives in the complex upper half-plane. This infinite product is the canonical \( q \)-Pochhammer symbol \( (q; q)_\infty \).

Alternatively, the exponential generating function form is
\[
\sum_{n=0}^{\infty} P_n(z) q^n = \exp\left(z \sum_{m=1}^\infty \frac{\sigma(m)}{m} q^m\right),
\]
where \( \sigma(m) = \sum_{d \mid m} d \) is the sum-of-divisors function [2601.11226; 2509.06123; 2601.07103].

The recursion is
\[
P_0(z) = 1, \qquad
P_n(z) = \frac{z}{n} \sum_{k=1}^{n} \sigma(k) P_{n-k}(z)
\]
for all \( n \ge 1 \) [1810.02226; 2601.11226].

D’Arcais polynomials are of degree \( n \), with rational coefficients and leading coefficient \( 1/n! \).

### Combinatorial Formulation

Via the Nekrasov–Okounkov hook-length formula,
\[
P_n(1-z) = \sum_{\lambda \vdash n} \prod_{h \in H(\lambda)} \left( \frac{h^2 + z}{h^2} \right ),
\]
where the sum runs over partitions \( \lambda \) of \( n \) and \( H(\lambda) \) denotes the multiset of hook-lengths of \( \lambda \) [1810.02226; 2509.06123].

### D’Arcais Numbers

The coefficients \( A(2,n,k) \) of \( z^k \) in \( P_n(z) \) are called D’Arcais numbers and can be extracted as [2601.07103]:
\[
A(2, n, k) = \frac{n!}{k!} [q^n]\left( -\ln((q; q)_\infty) \right)^k.
\]
This is a Bell transform of the abundancy index sequence \( \sigma(m)/m \).

## 2. Connections with Modular Forms and Partitions

The D’Arcais polynomials determine the coefficients in powers of the Dedekind eta function:
\[
\eta(\tau) = q^{1/24} \prod_{m=1}^\infty (1 - q^m),
\]
with the relation
\[
\eta(\tau)^{-z} = q^{-z/24} \prod_{m=1}^\infty (1 - q^m)^{-z} = q^{-z/24} \sum_{n=0}^\infty P_n(z) q^n.
\]
For positive integer \( z \), the values \( P_n(z) \) enumerate combinatorial quantities:
- \( P_n(1) = p(n) \) is the partition number [2601.11226].
- \( P_n(k) = p_k(n) \), the number of \( k \)-colored partitions [2601.11226].

Notably, the Ramanujan tau function arises as a specialization:
\[
\tau(n) = P_n(-24)
\]
[2004.06923].

## 3. Explicit Formulas, Recurrences, and Arrays

### Recursion and Triangular Arrays

Let \( A_{n,m} \) denote the coefficient of \( x^m \) in \( P_n(x) \), then
\[
A_{n,m} = \frac{1}{m!} [q^n] \left( \sum_{k=1}^\infty \sigma_1(k) q^k \right )^m
\]
or recursively,
\[
A_{n,m} = \frac{1}{m} \sum_{k=1}^{n-m+1} \sigma_1(k) A_{n-k, m-1}
\]
with \( A_{0,0} = 1 \), \( A_{n,0} = 0 \) if \( n \geq 1 \) [2010.05231].

### Partition and Hook Interpretations

There is a sum over partitions representation:
\[
P_n(x) = \sum_{\lambda \vdash n} \prod_{u \in \lambda} \left(1 + \frac{x}{h(u)^2} \right )
\]
[2010.07890].

### Bell Polynomial Formulas

For related generalizations, explicit Bell-polynomial expressions exist for certain extensions of D’Arcais polynomials [2004.06923].

## 4. Root Structure, Zero Distribution, and Non-Vanishing Results

### Root Distribution and Hurwitz Stability

Early conjectures posited that all roots of \( P_n(z) \) were real and negative. Counterexamples were found (e.g., \( n=10 \)), where nonreal roots emerge [1810.02226]. It is now conjectured and numerically substantiated that \( P_n(z) \) is Hurwitz-stable: all nontrivial roots satisfy \( \mathrm{Re}(\rho) < 0 \), and roots are simple.

### Non-Vanishing on Arithmetic Sets

Strong results have been established concerning non-vanishing at roots of unity and algebraic points. For instance,
- For any \( m \ge 3 \), \( P_n^g(\zeta_m) \neq 0 \) for prim. \( m \)-th root of unity \( \zeta_m \) [2509.06123; 2511.16276].
- Results extend to cyclotomic and quadratic integer translations under arithmetic constraints [2511.16276].
- The Lehmer conjecture that \( P_n^\sigma(24) \neq 0 \) for all \( n \) is equivalent to non-vanishing of Ramanujan's tau function [1810.02226; 2509.06123].

### Zero Location Transfer and Classical Orthogonals

A transfer mechanism links zero locations (e.g., between associated Laguerre and Chebyshev polynomials) and allows precise interval containment and root bounds for various classical polynomials [2304.02694]. For example, zeros of the associated Laguerre polynomials \( L_{n-1}^{(1)}(-z) \) admit explicit intervals determined by Chebyshev polynomial zeros.

## 5. Log-Concavity, Unimodality, and Large Deviations

### Log-Concavity and Unimodality

Extensive computational evidence indicates that the coefficient sequences of \( P_n(z) \) and their related forms are ultra-log-concave (i.e., \( b_k^2 \ge b_{k-1} b_{k+1} \)), and thus unimodal, for all checked degrees (up to at least \( n = 1000 \)). This property connects to the horizontal and vertical log-concavity of the associated triangular array of coefficients [1810.02226; 2010.05231].

### Large Deviation Estimates

A Bahadur–Rao type large deviation theorem holds for the normalized D’Arcais numbers:
\[
k_n! \frac{A(2,n,k_n)}{n!}
\]
in the regime \( k_n/n \to \kappa \in [0,1) \), governed by a rate function given by the Legendre–Fenchel transform of a function \( g \) defined via the abundancy index transform. Consequences include local log-concavity in the large-deviation regime and fine asymptotics for the distribution of the D’Arcais numbers [2601.07103].

## 6. Generalizations and Connections to Classical Polynomial Families

The methodology underlying D’Arcais polynomials extends to wide classes defined via recurrences with arbitrary weight functions \( g(n) \) and normalization sequences \( h(n) \). Notable examples include:
- Plane partitions: \( g(n) = \sum_{d|n} d^2 \)
- Chebyshev and associated Laguerre polynomials: identified via specific \( g, h \) selections
- Pochhammer and Hermite polynomials: realized in limiting or special parameter cases

These connections enable uniform approaches to analytic inequalities (e.g., Turán-type), zero location results, and log-concavity/unimodality phenomena across diverse enumerative and orthogonal polynomial families [2601.11226; 2304.02694; 2010.07890]. 

## 7. Open Problems and Ongoing Directions

Current prominent questions include:
- Proving analytic or bijective combinatorial proofs of coefficient ultra-log-concavity and full unimodality for all \( n \) [1810.02226; 2010.05231].
- Determining limiting distributions and possible universal root curves of \( P_n(z) \) as \( n \to \infty \) in the complex plane.
- Further developing algebraic number theory techniques (e.g., using the Dedekind–Kummer theorem) for full characterization of zero-sets for general \( g, h \).
- Establishing full non-vanishing ranges for coefficients for arithmetic and combinatorially significant specializations, incorporating generalized partition statistics and higher divisor sum functions.

These problems situate D’Arcais polynomials at the confluence of analytic, combinatorial, and arithmetic research, and underscore their deep significance in contemporary mathematics [1810.02226; 2601.11226; 2304.02694; 2511.16276].

Source: https://www.emergentmind.com/topics/d-arcais-polynomials