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