Building some symmetric Laguerre-Hahn functionals of
class two at most through 
the sum of symmetric functionals as
pseudofunctions with a Dirac measure at origin

Sghaier, M.; Alaya, J.

doi:https://doi.org/10.1155/IJMMS/2006/70835

International Journal of Mathematics and Mathematical Sciences

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 070835 | https://doi.org/10.1155/IJMMS/2006/70835

Building some symmetric Laguerre-Hahn functionals of class two at most through the sum of symmetric functionals as pseudofunctions with a Dirac measure at origin

M. Sghaier¹and J. Alaya²

Received16 May 2005

Revised04 Mar 2006

Accepted04 May 2006

Published06 Jul 2006

Abstract

We show that if v is a symmetric regular Laguerre-Hahn linear form (functional), then the linear form u defined by u=−λx−2v+δ0 is also regular and symmetric Laguerre-Hahn linear form for every complex λ except for a discrete set of numbers depending on v. We explicitly give the coefficients of the second-order recurrence relation, the structure relation of the orthogonal sequence associated with u, and the class of the linear form u knowing that of v. Finally, we apply the above results to the symmetric associated form of the first order for the classical polynomials.

References

J. Alaya and P. Maroni, “Some semi-classical and Laguerre-Hahn forms defined by pseudo-functions,” Methods and Applications of Analysis, vol. 3, no. 1, pp. 12–30, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. Alaya and P. Maroni, “Symmetric Laguerre-Hahn forms of class $s = 1$ ,” Integral Transforms and Special Functions, vol. 4, no. 4, pp. 301–320, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Beghdadi and P. Maroni, “On the inverse problem of the product of a semi-classical form by a polynomial,” Journal of Computational and Applied Mathematics, vol. 88, no. 2, pp. 377–399, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Branquinho and F. Marcellán, “Generating new classes of orthogonal polynomials,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 4, pp. 643–656, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. S. Chihara, “On co-recursive orthogonal polynomials,” Proceedings of the American Mathematical Society, vol. 8, no. 5, pp. 899–905, 1957.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
View at: Zentralblatt MATH | MathSciNet
E. B. Christoffel, “Über die Gaussische Quadratur und eine Verallgemeinerung derselben,” Journal für die reine und angewandte Mathematik, vol. 55, pp. 61–82, 1858.
View at: Google Scholar
J. Dini, Sur les formes linéaires et les polynômes orthogonaux de Laguerre-Hahn, Thése, Universite Pierre et Marie Curie, Paris, 1988.
J. Dini and P. Maroni, “Sur la multiplication d'une forme semi-classique par un polynôme,” Publ. Sem. Math. Univ. d'Antananarivo, vol. 3, pp. 76–89, 1989.
View at: Google Scholar
J. H. Lee and K. H. Kwon, “Division problem of moment functionals,” The Rocky Mountain Journal of Mathematics, vol. 32, no. 2, pp. 739–758, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Marcellán and E. Prianes, “Perturbations of Laguerre-Hahn linear functionals,” Journal of Computational and Applied Mathematics, vol. 105, no. 1-2, pp. 109–128, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Maroni, “Sur la décomposition quadratique d'une suite de polynômes orthogonaux. I,” Rivista di Matematica Pura ed Applicata, no. 6, pp. 19–53, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Maroni, “Sur la suite de polynômes orthogonaux associée à la forme $u = \delta _c + \lambda (x - c)^{ - 1} L$ ,” Periodica Mathematica Hungarica, vol. 21, no. 3, pp. 223–248, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Maroni, “Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques,” in Orthogonal Polynomials and Their Applications (Erice, 1990), C. Brezinski, L. Gori, and A. Ronveaux, Eds., vol. 9 of IMACS Ann. Comput. Appl. Math., pp. 95–130, Baltzer, Basel, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Maroni, “On a regular form defined by a pseudo-function,” Numerical Algorithms, vol. 11, no. 1, pp. 243–254, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. Maroni and I. Nicolau, “On the inverse problem of the product of a form by a polynomial: the cubic case,” Applied Numerical Mathematics, vol. 45, no. 4, pp. 419–451, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

198

Downloads

530

Citations