Dominant Zeros of Nekrasov–Okounkov Polynomials via Perron–Frobenius theory

Abstract We give an exact finite-dimensional Perron–Frobenius realization of the dominant zero of the Nekrasov–Okounkov polynomials $$\textrm{NO}_n(z)$$ NO n ( z ) . For a normalized positive sequence $$h=(h(n))_{n\ge 1}$$ h = ( h ( n ) ) n ≥ 1 with $$h(1)=1$$ h ( 1 ) = 1 , define $$\textrm{P}_0^h(z)=1$$ P 0 h ( z ) = 1 and, for $$n\ge 1$$ n ≥ 1 , $$\begin{aligned} \textrm{P}_n^h(z)=\frac{z}{h(n)}\sum _{k=1}^n \sigma (k)\textrm{P}_{n-k}^h(z), \end{aligned}$$ P n h ( z ) = z h ( n ) ∑ k = 1 n σ ( k ) P n - k h ( z ) , where $$\sigma (k)$$ σ ( k ) denotes the sum of divisors of k . The Nekrasov–Okounkov polynomials are obtained from the specialization $$h(n)=n$$ h ( n ) = n by the shif