SELF-RECIPROCAL POLYNOMIALS WITH RELATED MAXIMAL ZEROS

Abstract

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