Abstract
We carry out the complete description of the \(D_{w}\)-Laguerre–Hahn forms of class zero, where \(D_{w}\) is the divided difference operator. Essentially, four canonical cases appear. Some particular cases which refer to well-known orthogonal sequences are exhibited.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdelkarim, F.: Les polynômes \(D_{w}\)-classiques et les polynômes \(D_{w}\)-Laguerre–Hahn. Université de Tunis II, Thèse de Doctorat (1998)
Abdelkarim, F., Maroni, P.: The \(D_{w}\)-classical polynomials. Results Math. 32, 1–28 (1997)
Alaya, J., Maroni, P.: Symmetric Laguerre–Hahn forms of class \(s=1\). Integral Transforms Spec. Funct. 4, 301–320 (1996)
Bouakkaz, H.: Les polynômes orthogonaux de Laguerre–Hahn de classe zéro. Thèse de Doctorat. Université Pierre et Marie Curie, Paris (1990)
Bouakkaz, H., Maroni, P.: Description des polynômes orthogonaux de Laguerre-Hahn de classe zéro. In: Brezinski, C., Gori, L., Ronveaux, A. (eds.) Orthoggonals Polynomials and Their Applications. Annals of Computation and Applied Mathematics, vol. 9, pp. 189–194. J. C. Baltzer AG, Basel (1991)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)
Dini, J.: Sur les formes linéaires et polynômes oerthogonaux de Laguerre-Hahn. Université Pierre et Marie Curie, Paris VI, Thèse de Doctorat (1988)
Dzoumba, J.: Sur les polynômes de Laguerre–Hahn. Université Pierre et Marie Curie, Paris VI, Thèse de troisiéme cycle (1985)
Foupouagnigni, M., Marcellan, F.: Characterization of the \(D_{w}\)-Laguerre–Hahn Functionals. J. Differ. Equ. Appl. 8(8), 689–717 (2002)
Foupouagnigni, M.: Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator, fourth-order difference equation for the rth associated and the Laguerre–Freud equations for the recurrence coefficients. UniversitNationale du Bnin, Bnin (1998). Ph.D. Thesis
Guerfi, M.: Les polynômes de Laguerre–Hahn affines discrets, Thèse de troisième cycle. Univ. P. et M. Curie, Paris (1988)
Magnus, A.: Riccati acceleration of Jacobi continued fractions and Laguerre–Hahn orthogonal polynomials. In: Padé Approximation and its applications. Lecture Notes in Math, vol. 1071, 1984, pp. 213–230. Springer, Berlin (1983)
Maroni, P., Mejri, M.: The symmetric \(D_w\)-semiclassical orthogonal polynomials of class one. Numer. Algorithms 49, 251–282 (2008)
Maroni, P.: An introduction to second degree forms. Adv Comput Math 3, 59–88 (1995)
Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques. In: Brezinski, C., et al (eds.) Orthogonal polynomials and their applications. IMACS, Ann. Comput. Appl. Math. vol. 9, pp. 95–130. Baltzer, Basel (1991)
Zaatra, M.: Laguerre–Freud equations associated with the \(H_{q}\)-semi-classical forms of class one. Filomat 32(19), 6769–6787 (2018)
Acknowledgements
Thanks are due to the referee for his valuable comments and useful suggestions and for his careful reading of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sghaier, M., Zaatra, M. The \(D_{w}\)-Laguerre–Hahn Forms of Class Zero. Mediterr. J. Math. 17, 185 (2020). https://doi.org/10.1007/s00009-020-01630-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-01630-3