A family of self-inversive polynomials with concyclic zeros (Q1947460)

A polynomial \(p\) is said to be self-inversive if \(a_k=\zeta\, \overline{a_{n-k}}\) for \(0\leq k\leq n\) and \(|\zeta |=1\). For \(d\geq -\frac{1}{2}\), let \[ G(n,d,z):=\sum_{j=-n}^{n} \frac{\Gamma(d+j-1)\Gamma(d+j+1)}{(n-j)! (n+j)!}z^{n+j} \] be a family of self-inversive polynomials of degree \(2n\) with parameter \(d\). In the paper under review, the author investigates the distribution of zeros of \(G(n,d,z)\). In particular, the author shows that \(G(n,d,z)\) has all its zeros on the unit circle, if \(d> n-3/2\) (Theorem 1) or \(d=k-1/2\) for \(k=0,1,\dots,n-1\), (Corollary 2). In addition, the author calculates the discriminant of \(G(n,d,z)\) for \(d> n-3/2\) and the sum of the square distances between the zeros of \(G(n,d,z)\).