HAL: hal-00103572, version 1

DOI: 10.1016/j.cam.2004.05.025

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7th International Symposium on Orthogonal Polynomials, Special Functions and Applications, Copenhague : Danemark (2005)

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

Jacek Gilewicz 1, Elie Leopold 1, Galliano Valent 2

(2005)

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.

1:	Centre de Physique Théorique (CPT)
	CNRS : FR2291 – Université de Provence - Aix-Marseille I – Université de la Méditerranée - Aix-Marseille II – Université Sud Toulon Var
2:	Laboratoire de Physique Théorique et Hautes Energies (LPTHE)
	CNRS : UMR7589 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot

Subject

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Physics/Mathematical Physics

Keyword(s): orthogonal polynomials – indeterminate moment problems – Nevanlinna parametrization

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From: Galliano Valent <>

Submitted on: Wednesday, 4 October 2006 16:55:25

Updated on: Wednesday, 4 October 2006 17:27:27