A Rectangular Additive Convolution for Polynomials

Motivated by the study of singular values of random rectangular matrices, we define and study the rectangular additive convolution of polynomials with nonnegative real roots. Our definition directly generalizes the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava (2015), and our main theorem gives the corresponding generalization of the bound on the largest root from that paper. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges (2009). The proof is inductive, with the base case requiring new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest.