From Toeplitz matrix sequences to zero distribution of orthogonal polynomials

The authors are interested in sequences of orthogonal polynomials which are determined by a three term recurrence relation xpn,N(x)=an+1,Npn+1,N(x)+bn,Npn,N(x)+an,Npn−1,N(x), where the coefficients, and hence the orthogonal polynomials, depend not only on n but on an additional parameter N. In particular, it is assumed that N=N(n) and the ratio n/N(n)→t as t→∞. The main result describes the asymptotic behavior of the distribution of the zeros of pn,N as n/N→t. The authors take advantage of the fact that the zeros can be characterized as the eigenvalues of a Jacobi matrix, which can be approximated by certain matrices whose spectral distribution can be computed explicitly.