Dominant Zeros of Nekrasov--Okounkov Polynomials

We give an exact finite-dimensional Perron–Frobenius realization of the dominant zero of the Nekrasov–Okounkov polynomials ${\text{nop}}_n(z)$. For a normalized positive sequence $h=(h(n){)}_{n\ge 1}$ with $h(1)=1$, define ${\text{pol}}_0^h(z)=1$ and, for $n\ge 1$, $${\text{pol}}_n^h(z)=\frac{z}{h(n)}\sum _{k=1}^n\sigma (k){\text{pol}}_{n-k}^h(z),$$ where $\sigma (k)$ denotes the sum of divisors of $k$. The Nekrasov–Okounkov polynomials are obtained from the specialization $h(n)=n$ by the shift ${\text{nop}}_n(z)={\text{pol}}_n^h(z+1)$. We derive a Hessenberg determinant representation for ${\text{pol}}_n^h(z)$. After separating the trivial zero at the origin, the remaining zeros of ${\text{pol}}_n^h(-z)$ are identified with the eigenvalues of an explicit $(n-1)\times (n-1)$ nonnegative matrix $M_n^h$. We prove that $M_n^h$ is primitive and apply Perron–Frobenius theory to show that ${\text{pol}}_n^h(z)$ has a unique zero of maximal modulus; this zero is real, negative, and simple. As a consequence, the same property holds for the Nekrasov–Okounkov polynomials. We also prove strict monotonicity of the associated spectral radii.