SELF-RECIPROCAL POLYNOMIALS WITH RELATED MAXIMAL ZEROS

Abstract

For each real number n > 6, we prove that there is a sequence {p(k) (n, z))}(k = 1) (infinity) 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 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 qk(n)'s are polynomials in n defined by the severely nonlinear recurrence 4q2m-1(n) = q(2m-2)(2)(n) - (4n+1) Pi(m-2)(j=0) q(2)(j)(2)(n), 4q2m(n) = q(2m-2)(2)(n) - (n-2)(n-6) Pi(m-2)(j=0) q(2)(j)(2)(n) for m >= 1, with the usual empty product conventions, i.e., Pi(-1)(j = 0) b(j) = 1.