Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros (Q352739)

The authors study some results on pairs of cyclotomic polynomials whose zeros interlace on the unit circle and show that each of these pairs corresponds to a single polynomial from a special class of integer polynomials. In particular, they call a nonconstant polynomial with leading coefficient \(2\) \textit{disc-bionic} if all of its zeros lie in the open unit disc. They call a polynomial \(F\) \textit{\(2\)-reciprocal}, if all the coefficients of the difference between \(F\) and its reciprocal polynomial \(F^*\) are divisible by \(2\). In particular, in Corollary 2.2 they prove that (essentially) the list of all \(2\)-reciprocal disc-bionic polynomials consists of the infinite family \((2z^{n+m}-z^n-z^m)/(z-1)\), where \(m,n \in \mathbb{N}\), \(n \geq m\), \(\gcd(m,n)=1\), the infinite family \(2z^{n+m}+(-1)^{mn}z^n+z^m\), where \(m,n \in \mathbb{N}\), \(n > m\), \(\gcd(m,n)=1\), and some special \(26\) sporadic polynomials (which are extracted from an earlier work of Beukers and Heckman related to hypergeoemtric functions). Some applications to special Pisot numbers are also given.