Sums of two polynomials with each having real zeros symmetric with the other (Q698357)

Suppose that \(0<r_1\leq r_2\leq...\leq r_n.\) It is clear that the zeros of the polynomial \[ \prod_{k=1}^n(z-r_k)+\prod_{k=1}^n(z+r_k) \] lie on the imaginary axis. It is shown that each zero has multiplicity one, and the gaps between the zeros in the upper half-plane increase as one proceeds upward.