On skyburst polynomials and their zeros

Abstract

We consider polynomials orthogonal on the unit circle with respect to the complex-valued measure $z^{\omega -1}\mathrm{d}z$, where $\omega \in \mathbb{R}\setminus \{0\}$. We derive their explicit form, a generating function and several recurrence relations. These polynomials possess an intriguing pattern of zeros which, as $\omega$ varies, are reminiscent of a firework explosion. We prove this pattern in a rigorous manner.