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Laguerre–Freud equations associated with the D-Laguerre–Hahn forms of class one

Authors
  • Sghaier, Mabrouk
  • Zaatra, Mohamed
  • Khlifi, Achraf
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
Sep 11, 2018
Volume
10
Issue
4
Pages
395–411
Identifiers
DOI: 10.1515/apam-2018-0015
Source
De Gruyter
Keywords
License
Yellow

Abstract

We give the system of Laguerre–Freud equations for the recurrence coefficients β n {\beta_{n}} , γ n + 1 {\gamma_{n+1}} , n ≥ 0 {n\geq 0} , of orthogonal polynomials with respect to a D-Laguerre–Hahn form (linear functional) of class one. The system is solved in the case when β 0 = - θ 0 {\beta_{0}=-\theta_{0}} , β n + 1 = θ n - θ n + 1 {\beta_{n+1}=\theta_{n}-\theta_{n+1}} and γ n + 1 = - θ n 2 {\gamma_{n+1}=-\theta_{n}^{2}} with θ n ≠ 0 {\theta_{n}\neq 0} , n ≥ 0 {n\geq 0} . There are essentially three canonical cases.

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