ON SOME ROOT BEHAVIORS OF CERTAIN SUMS OF POLYNOMIALS

It is known that no two of the roots of the polynomial equation (1) Pi(n)(l=1) (x - r1) + Pi(n)(l=1) (x + r1) = 0, where 0 < r(1) <= r(2) <= ... <= r(n), can be equal and all of its roots lie on the imaginary axis. In this paper we show that for 0 < h < r(k), the roots of (x - r(k) + h) Pi(n)(l=1l not equal k) (x - r(1)) + (x + r(k) - h) Pi(n)(l=1l not equal k) (x + r(1)) = 0 and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis.