Some cubic birth and death processes and their related orthogonal polynomials

Jacek Gilewicz (1, 2) , Elie Leopold (1, 2) , Andreas Ruffing (3) , Galliano Valent (4)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 TUM - Zentrum Mathematik [Munchen]
4 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

The orthogonal polynomials with recurrence relation \[(\la_n+\mu_n-z)\,F_n(z)=\mu_{n+1}\,F_{n+1}(z)+\la_{n-1}\,F_{n-1}(z)\] with two kinds of cubic transition rates $\la_n$ and $\mu_n,$ corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices in the complex plane.

Mots clés: orthogonal polynomials; indeterminate moment problems; elliptic functions

Domaines: Physique mathématique [math-ph]

Jacek Gilewicz, Elie Leopold, Andreas Ruffing, Galliano Valent. Some cubic birth and death processes and their related orthogonal polynomials. Constructive Approximation, 2006, 24, pp.71-89. ⟨10.1007/s00365-005-0604-y⟩. ⟨hal-00014570⟩