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d'Arcais Polynomial

A d'Arcais polynomial, sometimes also called a Nekrasov-Okounkov polynomial, is a member of the sequence of polynomials defined by the generating function

sum_(n=0)^(infty)P_n^sigma(X)q^n=product_(m=1)^(infty)(1-q^m)^(-X)
(1)
=exp(Xsum_(j=1)^(infty)(sigma(j))/jq^j),
(2)

where sigma(n)=sigma_1(n) is the divisor function (d'Arcais 1913, Charlton et al. 2026). The first few are P_0^sigma(X)=1, P_1^sigma(X)=X, P_2^sigma(X)=X(X+3)/2, and P_3^sigma(X)=X(X+1)(X+8)/6.

Terminology in the literature is not uniform. Hong and Zhang (2021) use the name "Nekrasov-Okounkov polynomial" for the shifted polynomial Q_n(z)=P_n^sigma(z+1), while Charlton et al. (2026) use it for P_n^sigma(X) itself. In this entry, "d'Arcais polynomial" refers to the unshifted normalization defined above.

The identity P_n^sigma(-24)=tau(n+1) makes the assertion that the tau function never vanishes equivalent to the assertion that no d'Arcais polynomial has a root at X=-24.

Writing P_n^sigma(X)=sum_(k=0)^(n)p_n^sigma(k)X^k, it was conjectured that every d'Arcais polynomial is a logarithmically concave polynomial. Charlton et al. (2026), building on an asymptotic disproof by Starr (2026), showed that the first counterexample occurs at coefficient index k=2 for lambda=65214507758400, the 80th superabundant number, for which

[p_lambda^sigma(2)]^2<p_lambda^sigma(1)p_lambda^sigma(3).
(3)

See also

Dedekind Eta Function, Divisor Function, Logarithmically Concave Polynomial, Superabundant Number, Tau Function

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References

Charlton, S.; Heim, B.; and Stumpenhusen, J. "On the Smallest Counterexample to the Log-Concavity of the D'Arcais Polynomials." 18 Jun 2026. https://arxiv.org/abs/2606.09545.d'Arcais, F. "Développement en série." Intermédiaire Math. 20, 233-234, 1913.Hong, L. and Zhang, S. "Towards Heim and Neuhauser's Unimodality Conjecture on the Nekrasov-Okounkov Polynomials." Res. Number Theory 7, Paper No. 17, 11 pp., 2021. https://doi.org/10.1007/s40993-021-00244-2.Starr, S. "Asymptotics of the d'Arcais Numbers at Small k." 2 Feb 2026. https://arxiv.org/abs/2601.18599.

Cite this as:

Weisstein, Eric W. "d'Arcais Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/dArcaisPolynomial.html

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