Repeated polar differentiation of a polynomial sequence yields, in the large-degree limit, a polar free convolution power F^a_t, and these powers satisfy a Belinschi-Nica type commutation relation.
The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation.International Mathematics Research Notices, 2024(13):10189–10218,
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Asymptotic root distribution of polynomials under repeated polar differentiation
Repeated polar differentiation of a polynomial sequence yields, in the large-degree limit, a polar free convolution power F^a_t, and these powers satisfy a Belinschi-Nica type commutation relation.