Let q be a power of 2 such that \(q+1\) is a Fermat prime. In this paper, we establish connections between the stability of polynomials of the form \(f(x)=b^q(x+a)^{q+1}+x\) over odd-degree extensions of \(\mathbb {F}_{q}\) and the properties of two sequences \((\beta _n)_{n\ge 0}\) and \((A_r(x))_{r\ge 0}\) , where \((\beta _n)_{n\ge 0}\) , the terms of which are roots of iterates of f(x), is constructed through Capelli’s Lemma, and the polynomial sequence \((A_r(x))_{r\ge 0}\) is defined by the initial values \(A_0(x)=0,A_1(x)=1\) and the recurrence relation \(A_{r+2}(x)=A_{r+1}(x)+x^{q^r}A_r(x)\) for \(r\ge 0\) . Through studying the properties of \((A_r(\beta _n))_{r\ge 0}\) , this paper extends our previous results on stable cubic polynomials over extensions of \(\mathbb {F}_{2}\) to stability criteria for polynomials of the above-mentioned form whose degrees are arbitrary Fermat primes.
