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

Jacek Gilewicz (1, 2) , Elie Leopold (1, 2) , Galliano Valent (3)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

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.

Mots clés: orthogonal polynomials; indeterminate moment problems; Nevanlinna parametrization

Domaines: Physique mathématique [math-ph]

Jacek Gilewicz, Elie Leopold, Galliano Valent. New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes. 7th International Symposium on Orthogonal Polynomials, Special Functions and Applications, 2005, Copenhague, Denmark. pp.235-245, ⟨10.1016/j.cam.2004.05.025⟩. ⟨hal-00103572⟩