Class of symmetric \(H_{\sqrt{q}}\)-Laguerre-Hahn linear forms

Sobhi Jbeli

doi:10.56754/0719-0646.2802.409

DOI:

https://doi.org/10.56754/0719-0646.2802.409

Abstract

The aim of this paper is to study the symmetrized form associated with a \(H_q\)-Laguerre-Hahn form, where \(H_q\) is the \(q\)-derivative operator. Given a \(H_q\)-Laguerre-Hahn form \(u\) of class \(s\), it is shown that its symmetrized form \(w\) is \(H_{\sqrt{q}}\)-Laguerre Hahn of class \(\tilde{s}\leq 2s+3\). We give the \(\sqrt{q}\)-Riccati equation satisfied by the Stieltjes formal series \(S(w)\) as well as a complete discussion of the class \(\tilde{s}\).

As an application of this work, we generate two examples of symmetric \(H_{\sqrt{q}}\)-Laguerre-Hahn orthogonal polynomials of class two and three.

Keywords

Orthogonal q-polynomials , q-derivative operator , q-difference equation , q-Riccati equation , Hq-Laguerre-Hahn character , quadratic decomposition

Mathematics Subject Classification:

33C45 , 42C05

Sobhi Jbeli University of Jendouba, Higher Institute of Computer Science of Kef, 5 Saleh Ayech Street, 7100, Kef. 2Faculty of Sciences of Tunis, El Manar University Campus, Tunis, 2092, Tunisia - Research laboratory: Mathematical modeling, Harmonic analysis, and potential theory. LR18ES09, Tunis, Tunisia. https://orcid.org/0000-0001-8540-8053

Pages: 409-427
Date Published: 2026-05-31
Vol. 28 No. 2 (2026)

J. Alaya and P. Maroni, “Symmetric Laguerre-Hahn forms of class s= 1,” Integral Transforms Spec. Funct., vol. 4, no. 4, pp. 301–320, 1996, doi: 10.1080/10652469608819117.

J. Arvesú, J. Atia, and F. Marcellán, “On semiclassical linear functionals: The symmetric companion,” Commun. Anal. Theory Contin. Fract., vol. 10, pp. 13–29, 2002.

H. Bouakkaz and P. Maroni, “Description des polynômes orthogonaux de Laguerre-Hahn de classe zéro,” in Orthogonal polynomials and their applications. Proceedings of the third international symposium held in Erice, Italy, June 1-8, 1990. Basel: J. C. Baltzer, 1991, pp. 189–194.

A. Branquinho and F. Marcellán, “Generating new classes of orthogonal polynomials,” Int. J. Math. Math. Sci., vol. 19, no. 4, pp. 643–656, 1996, doi: 10.1155/S0161171296000919.

T. S. Chihara, An introduction to orthogonal polynomials, ser. Math. Appl., Gordon Breach Sci. Publ. Gordon & Breach Science Publishers, New York, NY, 1978, vol. 13.

H. Dueñas and L. E. Garza, “Perturbations of Laguerre-Hahn class linear functionals by Dirac delta derivatives,” Bol. Mat. (N.S.), vol. 19, no. 1, pp. 65–90, 2012.

H. Dueñas, F. Marcellán, and E. Prianes, “Perturbations of Laguerre-Hahn functional: modification by the derivative of a Dirac delta,” Integral Transforms Spec. Funct., vol. 20, no. 1, pp. 59–77, 2009, doi: 10.1080/10652460802493177.

G. Gasper and M. Rahman, Basic hypergeometric series, 2nd ed., ser. Encycl. Math. Appl. Cambridge: Cambridge University Press, 2004, vol. 96.

A. Ghressi, L. Khériji, and M. I. Tounsi, “An introduction to the q-Laguerre-Hahn orthogonal q-polynomials,” SIGMA, Symmetry Integrability Geom. Methods Appl., vol. 7, 2011, Art. ID 092, doi: 10.3842/SIGMA.2011.092.

W. Hahn, “On orthogonal polynomials satisfying q-difference equations,” Math. Nachr., vol. 2, pp. 4–34, 1949, doi: 10.1002/mana.19490020103.

M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable., ser. Encycl. Math. Appl. Cambridge: Cambridge University Press, 2005, vol. 98.

S. Jbeli and L. Khériji, “On the addition of a Dirac mass to a q-Laguerre-Hahn form.” Transylvanian Journal of Mathematics & Mechanics, vol. 14, no. 1, pp. 53–62, 2022.

S. Jbeli and L. Khériji, “On some perturbed q-Laguerre-Hahn orthogonal q-polynomials,” Period. Math. Hung., vol. 86, no. 1, pp. 115–138, 2023, doi: 10.1007/s10998-022-00463-9.

S. Jbeli, “Description of the symmetric Hq-Laguerre-Hahn orthogonal q-polynomials of class one,” Period. Math. Hung., vol. 89, no. 1, pp. 86–106, 2024, doi: 10.1007/s10998-024-00574-5.

S. Jbeli and L. Khériji, “Characterization of symmetrical Hq-Laguerre-Hahn orthogonal polynomials of class zero,” Filomat, vol. 38, no. 24, pp. 8349–8365, 2024, doi: 10.2298/FIL2424349J.

L. Khériji and P. Maroni, “The Hq-classical orthogonal polynomials,” Acta Appl. Math., vol. 71, no. 1, pp. 49–115, 2002, doi: 10.1023/A:1014597619994.

L. Kheriji, “An introduction to the Hq-semiclassical orthogonal polynomials,” Methods Appl. Anal., vol. 10, no. 3, pp. 387–412, 2003, doi: 10.4310/MAA.2003.v10.n3.a5.

L. Khériji, “On the Al-Salam-Carlitz orthogonal q-polynomials,” Quaest. Math., vol. 35, no. 2, pp. 229–234, 2012, doi: 10.2989/16073606.2012.697263.

F. Marcellán and J. Petronilho, “Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings,” Linear Algebra Appl., vol. 260, pp. 169–208, 1997, doi: 10.1016/S0024-3795(97)80009-2.

P. Maroni, “Une théorie algebrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques,” in Orthogonal polynomials and their applications. Proceedings of the third international symposium held in Erice, Italy, June 1-8, 1990. Basel: J.C.Baltzer, 1991, pp. 95–130.

P. Maroni and M. Mejri, “The I(q,ω) classical orthogonal polynomials,” Appl. Numer. Math., vol. 43, no. 4, pp. 423–458, 2002, doi: 10.1016/S0168-9274(01)00180-5.

M. Sghaier, M. Zaatra, and M. Mechri, “The symmetric Hq-Laguerre-Hahn orthogonal polynomials of class zero,” Azerb. J. Math., vol. 13, no. 1, pp. 34–50, 2023.

M. Sghaier and M. Zaatra, “On Laguerre-Hahn linear functionals: the symmetric companion,” Adv. Pure Appl. Math., vol. 1, no. 3, pp. 345–358, 2010, doi: 10.1515/APAM.2010.023.

M. I. Tounsi, I. Ben Salah, and L. Khriji, “On the symmetric Hq-semiclassical polynomial sequences of even class. Some examples from the class two,” Mediterr. J. Math., vol. 10, no. 3, pp. 1293–1316, 2013, doi: 10.1007/s00009-012-0236-y.