#### Vol. 14, No. 2, 2021

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Zeros of complex random polynomials spanned by Bergman polynomials

### Marianela Landi, Kayla Johnson, Garrett Moseley and Aaron Yeager

Vol. 14 (2021), No. 2, 271–281
##### Abstract

We study the expected number of zeros of

${P}_{n}\left(z\right)=\sum {k=0}^{n}{\eta }_{k}{p}_{k}\left(z\right),$

where $\left\{{\eta }_{k}\right\}$ are complex-valued independent and identically distributed standard Gaussian random variables, and $\left\{{p}_{k}\left(z\right)\right\}$ are polynomials orthogonal on the unit disk. When ${p}_{k}\left(z\right)=\sqrt{\left(k+1\right)∕\pi }{z}^{k}$, $k\in \left\{0,1,\dots ,n\right\}$, we give an explicit formula for the expected number of zeros of ${P}_{n}\left(z\right)$ in a disk of radius $r\in \left(0,1\right)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2n∕3$. Generalizing our basis functions $\left\{{p}_{k}\left(z\right)\right\}$ to be regular in the sense of Ullman–Stahl–Totik and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros in a disk of radius $r\in \left(0,1\right)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2n∕3$.

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##### Keywords
random polynomials, Bergman polynomials, Ullman–Stahl–Totik regular
##### Mathematical Subject Classification
Primary: 30C15, 30E15, 30C40
Secondary: 60B99
##### Milestones
Received: 7 July 2020
Revised: 11 January 2021
Accepted: 12 January 2021
Published: 6 April 2021

Communicated by John C. Wierman
##### Authors
 Marianela Landi Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States Kayla Johnson Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States Garrett Moseley Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States Aaron Yeager Department of Mathematics and Data Science College of Coastal Georgia Brunswick, GA United States