Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

We analyze polynomials Pn that are biorthogonal to exponentials {e−σn,j }j=1, in the sense that ∫ ∞ 0 Pn(x)e −σn,j x dx = 0, 1 ≤ j ≤ n. Here α >−1. We show that the zero distribution of Pn as n→∞ is closely related to that of the associated exponent polynomial Qn(y)= n ∏ j=1 (y + 1/σn,j )= n ∑ j=0 qn,j y j . More precisely, we show that the zero counting measures of {Pn(−4nx)}∞n=1 converge weakly if and only if the zero counting measures of {Qn}∞n=1 converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial n ∑ j=0 qn,j n,j x j , under appropriate assumptions on { n,j }. Communicated by Serguei Denissov. D.S. Lubinsky ( ) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA e-mail: [email protected] A. Sidi Department of Computer Science, Technion-Israel Institute of Technology, Haifa 32000, Israel e-mail: [email protected] 344 Constr Approx (2008) 28: 343–371