Geodesic
An Introduction to the $q$-Laguerre–Hahn Orthogonal $q$-Polynomials
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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Orthogonal $q$-polynomials associated with $q$-Laguerre–Hahn form will be studied as a generalization of the $q$-semiclassical forms via a suitable $q$-difference equation. The concept of class and a criterion to determinate it will be given. The $q$-Riccati equation satisfied by the corresponding formal Stieltjes series is obtained. Also, the structure relation is established. Some illustrative examples are highlighted.
Keywords: orthogonal $q$-polynomials; $q$-Laguerre–Hahn form; $q$-difference operator; $q$-difference equation; $q$-Riccati equation. 
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Abdallah Ghressi; Lotfi Khériji; Mohamed Ihsen Tounsi. An Introduction to the $q$-Laguerre–Hahn Orthogonal $q$-Polynomials. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a91/

[1] Alaya J., Maroni P., “Symmetric Laguerre–Hahn forms of class $s=1$”, Integral Transform. Spec. Funct., 2 (1996), 301–320 | DOI | MR

[2] Alaya J., Maroni P., “Some semi-classical and Laguerre–Hahn forms defined by pseudo-functions”, Methods Appl. Anal., 3 (1996), 12–30 | MR | Zbl

[3] Álvarez-Nodarse R., Medem J.C., “$q$-classical polynomials and the $q$-Askey and Nikiforov–Uvarov tableaux”, J. Comput. Appl. Math., 135 (2001), 197–223 | DOI | MR

[4] Bangerezako G., “The fourth order difference equation for the Laguerre–Hahn polynomials orthogonal on special non-uniform lattices”, Ramanujan J., 5 (2001), 167–181 | DOI | MR | Zbl

[5] Bangerezako G., An introduction to $q$-difference equations, Bujumbura, 2008

[6] Bouakkaz H., Maroni P., “Description des polynômes de Laguerre–Hahn de classe zéro”, Orthogonal Polynomials and Their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math., 9, Baltzer, Basel, 1991, 189–194 | MR | Zbl

[7] Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, 13, Gordon and Breach Science Publishers, New York – London – Paris, 1978 | MR | Zbl

[8] Dini J., Sur les formes linéaires et polynômes oerthogonaux de Laguerre–Hahn, Thèse de Doctorat, Université Pierre et Marie Curie, Paris VI, 1988 | Zbl

[9] Dini J., Maroni P., Ronveaux A., “Sur une perturbation de la récurrence vérifiée par une suite de polynômes orthogonaux”, Portugal. Math., 46 (1989), 269–282 | MR | Zbl

[10] Dzoumba J., Sur les polynômes de Laguerre–Hahn, Thèse de 3 ème cycle, Université Pierre et Marie Curie, Paris VI, 1985

[11] Foupouagnigni M., Ronveaux A., Koepf W., “Fourth order $q$-difference equation for the first associated of the $q$-classical orthogonal polynomials”, J. Comput. Appl. Math., 101 (1999), 231–236 | DOI | MR | Zbl

[12] Foupouagnigni M., Ronveaux A., “Difference equation for the co-recursive $r$th associated orthogonal polynomials of the $D_q$-Laguerre–Hahn class”, J. Comput. Appl. Math., 153 (2003), 213–223 | DOI | MR | Zbl

[13] Foupouagnigni M., Marcellán F., “Characterization of the $D_\omega$-Laguerre–Hahn functionals”, J. Difference Equ. Appl., 8 (2002), 689–717 | DOI | MR | Zbl

[14] Ghressi A., Khériji L., “The symmetrical $H_{q}$-semiclassical orthogonal polynomials of class one”, SIGMA, 5 (2009), 076, 22 pp. ; arXiv: 0907.3851 | DOI | MR | Zbl

[15] Guerfi M., Les polynômes de Laguerre–Hahn affines discrets, Thèse de troisième cycle, Univ. P. et M. Curie, Paris, 1988

[16] Khériji L., Maroni P., “The $H_{q}$-classical orthogonal polynomials”, Acta. Appl. Math., 71 (2002), 49–115 | DOI | MR

[17] Khériji L., “An introduction to the $H_{q}$-semiclassical orthogonal polynomials”, Methods Appl. Anal., 10 (2003), 387–411 | MR

[18] Magnus A., “Riccati acceleration of Jacobi continued fractions and Laguerre–Hahn orthogonal polynomials”, Padé Approximation and its Applications (Bad Honnef, 1983), Lecture Notes in Math., 1071, Springer, Berlin, 1984, 213–230 | MR

[19] Marcellán F., Salto M., “Discrete semiclassical orthogonal polynomials”, J. Difference. Equ. Appl., 4 (1998), 463–496 | DOI | MR | Zbl

[20] Maroni P., “Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classique”, Orthogonal Polynomials and their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math., 9, Baltzer, Basel, 1991, 95–130 | MR | Zbl

[21] Medem J.C., Álvarez-Nodarse R., Marcellán F., “On the $q$-polynomials: a distributional study”, J. Comput. Appl. Math., 135 (2001), 157–196 | DOI | MR | Zbl