On skyburst polynomials and their zeros

María José Cantero; Arieh Iserles

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arXiv · 2308.06059

On skyburst polynomials and their zeros

María José Cantero·Arieh Iserles

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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.

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María José Cantero, Arieh Iserles. 2023-08-11. On skyburst polynomials and their zeros. https://arxiv.org/abs/2308.06059

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