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 dd goes from −1/2−1/2 to ∞∞, the polynomial goes from a constant multiple of (z−1)2n(z−1)2n to a constant multiple of (z+1)2n(z+1)2n with the polynomial ∑k=02nzk when d=nd=n. Also, we compute the discriminants and the squared distance sums of their zeros.