Some properties of classes of real self-reciprocal polynomials

The purpose of this paper is twofold. Firstly we investigate the distribution, simplicity and monotonicity of the zeros around the unit circle and real line of the real self-reciprocal polynomials Rn(λ)(z)=1+λ(z+z2+⋯+zn−1)+zn, n≥2n≥2 and λ∈Rλ∈R. Secondly, as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self-reciprocal polynomials Sn(λ)(z)=∑k=0nsn,k(λ)zk, n≥2n≥2, with sn,0(λ)=sn,n(λ)=1, sn,n−k(λ)=sn,k(λ)=1+kλ, k=1,2,…,⌊n/2⌋k=1,2,…,⌊n/2⌋ when n   is odd, and sn,n−k(λ)=sn,k(λ)=1+kλ, k=1,2,…,n/2−1k=1,2,…,n/2−1, sn,n/2(λ)=(n/2)λ when n is even, lie on the unit circle, solving then an open problem given by Kim and Park in 2008.