New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes

摘要

The orthogonal polynomials with recurrence relation(λn+μn-z)Fn(z)=μn+1Fn+1(z)+λn-1Fn-1(z)and the three kinds of cubic transition ratesλn=(3n+1)2(3n+2),μn=(3n-1)(3n)2,λn=(3n+2)2(3n+3),μn=3n(3n+1)2,λn=(3n+1)(3n+2)2,μn=(3n)2(3n+1),correspond to indeterminate Stieltjes moment problems. It follows that the polynomials Fn(z) have infinitely many orthogonality measures, whose Stieltjes transform is obtained from their Nevanlinna matrix, a 2×2 matrix of entire functions. We present the full Nevanlinna matrix for these three classes of polynomials and we discuss its growth at infinity and the asymptotic behaviour of the mass points for the Nevanlinna extremal measures.