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

We give an exact finite-dimensional Perron–Frobenius realization of the dominant zero of the Nekrasov–Okounkov polynomials NOₙ(z) NO n ( z ) . For a normalized positive sequence h=(h(n))n≥ 1 h = ( h ( n ) ) n ≥ 1 with $$h(1)=1$$ h ( 1 ) = 1 , define P₀ʰ(z)=1 P 0 h ( z ) = 1 and, for n≥ 1 n ≥ 1 , aligned Pₙʰ(z)=z/h(n)∑ ₖ₌₁ⁿ σ (k)Pₙ₋ₖʰ(z), aligned P n h ( z ) = z h ( n ) ∑ k = 1 n σ ( k ) P n - k h ( z ) , where σ (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