Zero distribution of composite polynomials and polynomials biorthogonal to exponentials (Q1039889)

Biorthogonal polynomials in continuous or discrete settings have been studied by many authors. In the present paper the authors deal with monic polynomials \(P_n\) of degree \(n\) determined by the biorthogonality relations \[ \int_0^\infty P_n(x)x^{\alpha}e^{-\sigma_{n,j}x}\,dx=0, \quad 1\leq j\leq n, \] where \(\alpha>-1\) and \(\{\sigma_{n,j}\}_{j=1}^n\) is an array of distinct positive numbers. They prove that \(P_n\) exists, is unique, has \(n\) simple positive zeros, and is given explicitly by the composition of the associated exponent polynomial \[ Q_n(x)=\prod_{j=1}^n (x+\sigma_{n,j}^{-1}) \] and the Laguerre polynomial \(L_n^{(\alpha)}\). The authors study the zero distribution of \(\{P_n\}\) and show that the zero counting measures \(\mu_n\) of \(\{P_n(-4nx)\}\) and the zero counting measures \(\nu_n\) of \(\{Q_n\}\) have weak limits simultaneously, and find the relation between the corresponding limits.