Jaegug Bae and Seon-Hong Kim Korea Maritime University, Sookmyung Women's University

Abstract : For each real number $n>6$, we prove that there is a sequence $\{ p_k(n,z)\}_{k=1}^{\infty}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $\, p_{k+1}(n,z) $ has the largest (in modulus) zero $\alpha \beta$ where $\alpha$ and $\beta$ are the first and the second largest (in modulus) zeros of $p_{k}(n,z)$, respectively. One such sequence is given by $p_{k}(n,z)$ so that $$ p_{k}(n,z)= z^4 -q_{k-1}(n)\,z^3+(q_k(n)+2)\,z^2-q_{k-1}(n)\,z+1, $$ where $q_0(n)=1$ and other $q_k(n)$'s are polynomials in $n$ defined by the severely nonlinear recurrence \begin{align*} 4q_{2m-1}(n)&=q_{2m-2}^{2}(n)-(4n+1)\prod_{j=0}^{m-2}q_{2j}^{2}(n),\\ 4q_{2m}(n)&=q_{2m-1}^{2}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}q_{2j+1}^{2}(n) \end{align*} for $m\ge1$, with the usual empty product conventions, i.e., $ \prod_{j=0}^{-1}b_j=1$.