On skyburst polynomials and their zeros

TL;DR

This paper studies a special class of orthogonal polynomials on the unit circle with complex weights, deriving explicit formulas and analyzing their zeros, which form a firework-like pattern as a parameter varies.

Contribution

It provides explicit forms, generating functions, recurrence relations, and a rigorous proof of the zero pattern for skyburst polynomials with complex weights.

Findings

Zeros form a firework explosion pattern as parameter varies

Explicit formulas and recurrence relations derived

Rigorous proof of zero distribution pattern

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.