ON ZERO DISTRIBUTIONS OF SOME SELF-RECIPROCAL POLYNOMIALS WITH REAL COEFFICIENTS

Abstract

If q(z) is a polynomial of degree n with all zeros in the unit circle, then the self-reciprocal polynomial q(z) + xn q(1/z) has all its zeros on the unit circle. One might naturally ask: where are the zeros of q(z) + xn q(1/z) located if q(z) has different zero distribution from the unit circle? In this paper, we study this question when q(z)=(z-1)n-k (z-1-c1 )\cdots(z-1-ck ) +(z+1)n-k (z+1+c1)\cdots(z+1+ck), where cj>0 for each j, and q(z) is a `'zeros dragged' polynomial from (z-1)n+(z+1)n whose all zeros lie on the imaginary axis.