SELF-RECIPROCAL POLYNOMIALS WITH RELATED MAXIMAL ZEROS

Title & Authors
SELF-RECIPROCAL POLYNOMIALS WITH RELATED MAXIMAL ZEROS
Bae, Jaegug; Kim, Seon-Hong;

Abstract
For each real number $\small{n}$ > 6, we prove that there is a sequence $\small{\{pk(n,z)\}^{\infty}_{k=1}}$ of fourth degree self-reciprocal polynomials such that the zeros of $\small{p_k(n,z)}$ are all simple and real, and every $\small{p_{k+1}(n,z)}$ has the largest (in modulus) zero $\small{{\alpha}{\beta}}$ where $\small{{\alpha}}$ and $\small{{\beta}}$ are the first and the second largest (in modulus) zeros of $\small{p_k(n,z)}$, respectively. One such sequence is given by $\small{p_k(n,z)}$ so that $\small{p_k(n,z)=z^4-q_{k-1}(n)z^3+(q_k(n)+2)z^2-q_{k-1}(n)z+1}$, where $\small{q_0(n)=1}$ and other $\small{q_k(n)^{\prime}s}$ are polynomials in n defined by the severely nonlinear recurrence $\small{4q_{2m-1}(n)=q^2_{2m-2}(n)-(4n+1)\prod_{j=0}^{m-2}\;q^2_{2j}(n),\\4q_{2m}(n)=q^2_{2m-1}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}\;q^2_{2j+1}(n)}$ for $\small{m{\geq}1}$, with the usual empty product conventions, i.e., $\small{{\prod}_{j=0}^{-1}\;b_j=1}$.
Keywords
self-reciprocal polynomials;polynomials;sequences;
Language
English
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References
1.
T. Sheil-Small, Complex Polynomials, Cambridge Studies in Advaced Mathematics 73, Cambridge University Press, Cambridge, 2002.