A rectangular additive convolution for polynomials

Understand

We define the rectangular additive convolution of polynomials with nonnegative real roots as a generalization of the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava.

• We then prove a sliding bound on the largest root of this convolution.
• The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges.
• The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest.