Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

We analyze polynomials P n that are biorthogonal to exponentials $$\int_{0}^{\infty }P_{n}(x)e^{-\sigma _{n,j}x}x^{\alpha }\,dx=0,\quad 1\leq j\leq n.$$Here α>−1. We show that the zero distribution of P n as n→∞ is closely related to that of the associated exponent polynomial$$Q_{n}(y)=\prod\limits_{j=1}^{n}(y+1/\sigma _{n,j})=\sum_{j=0}^{n}q_{n,j}y^{j}.$$More precisely, we show that the zero counting measures of {P n (−4nx)} n=1∞ converge weakly if and only if the zero counting measures of {Q n } n=1∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial$$\sum_{j=0}^{n}q_{n,j}\Delta _{n,j}x^{j},$$under appropriate assumptions on {Δ n,j }.