A rectangular additive convolution for polynomials

Motivated by the study of singular values random rectangular matrices, we define and additive convolution polynomials with nonnegative real roots. Our definition directly generalizes asymmetric introduced Marcus, Spielman Srivastava (2015), our main theorem gives corresponding generalization bound on largest root from that paper. The tool used in analysis is a differential operator derived "rectangular Cauchy transform" Benaych-Georges (2009). proof inductive, base case requiring new nonasymptotic transform Gegenbauer which may be independent interest.Mathematics Subject Classifications: 26C10, 33C45