New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes
Résumé
The orthogonal polynomials with recurrence relation \[(\la_n+\mu_n-z)\,F_n(z)=\mu_{n+1}\,F_{n+1}(z)+\la_{n-1}\,F_{n-1}(z)\] and the three kinds of cubic transition rates \[\left\{\barr{ll} \la_n=(3n+1)^2(3n+2), & \qq\mu_n=(3n-1)(3n)^2,\\[4mm] \la_n=(3n+2)^2(3n+3), & \qq\mu_n=3n(3n+1)^2,\\[4mm] \la_n=(3n+1)(3n+2)^2, & \qq\mu_n=(3n)^2(3n+1),\earr\right.\] correspond to indeterminate Stieltjes moment problems. It follows that the polynomials $\,F_n(z)\,$ have infinitely many orthogonality measures, whose Stieltjes transform is obtained from their Nevanlinna matrix, a $2\times 2$ matrix of entire functions. We present the full Nevanlinna matrix for these three classes of polynomials and we discuss its growthat infinity and the asymptotic behaviour of the spectra of the Nevanlinna extremal measures.
Loading...