Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights

TL;DR

This paper derives the asymptotic expected number of real zeros for certain random polynomials formed from orthogonal polynomials associated with radial weights, expanding understanding of their zero distribution.

Contribution

It provides the first asymptotic analysis of the expected number of real roots for these specific random orthogonal polynomial combinations.

Findings

Asymptotic expected number of real zeros derived

Results apply to polynomials with radial weight functions

Extends previous work on random polynomial zeros

Abstract

In this note, we obtain asymptotic expected number of real zeros for random polynomials of the form $$f_n(z)=\sum_{j=0}^na^n_jc^n_jz^j$$ where $a^n_j$ are independent and identically distributed real random variables with bounded $(2+\delta)$th absolute moment and the deterministic numbers $c^n_j$ are normalizing constants for the monomials $z^j$ within a weighted $L^2$-space induced by a radial weight function satisfying suitable smoothness and growth conditions.