Some properties of classes of real self-reciprocal polynomials - Botta, Vanessa [UNESP]; Bracciali, Cleonice F. [UNESP]; Pereira, Junior A. [UNESP] (My University, 2016)
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≥. 2 and λ∈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≥. 2, with sn,0(λ)=sn, n(λ)=1, sn, n-k(λ)=sn, k(λ)=1+kλ, k=1,2,.,⌊n/2⌋ when n is odd, and sn, n-k(λ)=sn, k(λ)=1+kλ, k=. 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.
Keywords: Interlacing; Monotonicity; Self-reciprocal polynomials; Unit circle; Zeros