A family of self-inversive polynomials with concyclic zeros

We give a one-parameter family of self-inversive polynomials associated with Jacobi polynomials that has all zeros on the unit circle. As the parameter d goes from − 1 / 2 to ∞ , the polynomial goes from a constant multiple of ( z − 1 ) 2 n to a constant multiple of ( z + 1 ) 2 n with the polynomial ∑ k = 0 2 n z k when d = n . Also, we compute the discriminants and the squared distance sums of their zeros.