On the limit distributions of the zeros of Jonquière polynomials and generalized classical orthogonal polynomials (Q1890555)

Some sequences of polynomials with real zeros are investigated, with emphasis on the asymptotic distribution of the zeros. First the Jonquière polynomials \(J_k (z)\) are investigated, which are defined by \[ {{J_k (z)} \over {(1- z)^{k+ 1}}}= \sum_{n=0}^\infty n^k z^n. \] The zero distribution was found earlier by Gawronski and Stadtmüller, but now the authors consider a generalization of the form \[ {{J_k (z)} \over {\prod_{j= 1}^{k+ 1} (1- ze^{\beta_{k+ 1,j}})}}= \sum_{n= 0}^\infty E_k (n) z^n, \] where the \(E_k\) are exponential polynomials satisfying \(Q_{k+ 1} (d/dx) E_k (x) =0\), with \(Q_{k+1}\) a polynomial of degree \(k+1\) with zeros \(\beta_{k+ 1,j}\), \(j=1, 2, \dots, k+1\). If the asymptotic distribution \(\mu\) of the zeros of \(Q_{k+1}\) for \(k\to \infty\) is known, then the authors give the asymptotic distribution of the zeros of \(J_k\) in terms of \(\mu\). In the second part of the paper, the authors give the asymptotic zero distribution of polynomials which are a solution of a second order differential equation for which the coefficients have prescribed asymptotic behaviour as the degree of the polynomials tends to infinity. This case covers Jacobi polynomials \(P_n^{(\alpha_n, \beta_n)}\) where the parameters may depend on the degree \(n\). Also Laguerre polynomials \(L_n^{( \alpha_n)}\) and generalized Hermite polynomials \(H_n^{( \alpha_n)}\), with weight function \(|x|^{2 \alpha_n} e^{- x^2}\) on \((- \infty, \infty)\), are considered.