Some semiclassical orthogonal polynomials of class one
Eurasian mathematical journal, Tome 2 (2011) no. 2, pp. 108-128.

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In this work, we will solve the Laguerre-Freud equations for the recurrence coefficients of the semiclassical orthogonal polynomials of class one in a particular case. The integral representation and, as a consequence, the moments of the corresponding form are obtained. Furthermore, both the characteristic elements of the structure relation and of the second-order differential equation are explicitly given.
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P. Maroni; M. Mejri. Some semiclassical orthogonal polynomials of class one. Eurasian mathematical journal, Tome 2 (2011) no. 2, pp. 108-128. http://geodesic.mathdoc.fr/item/EMJ_2011_2_2_a5/

[1] J. Alaya, P. Maroni, “Symmetric Laguerre-Hahn forms of class $s=1$”, Int. Transf. Spec. Funct., 4 (1996), 301–320 | DOI | MR | Zbl

[2] M. J. Atia, F. Marcellan, I. A. Rocha, “On semiclassical orthogonal polynomials: a quasi-definite functional of class 1”, Facta Universitatis. Ser. Math. Inform., 17 (2002), 13–34 | MR | Zbl

[3] M. Bachène, Les polynômes semi-classiques de classe zéro et de classe un, Thesis of third cycle, Université Pierre et Marie Curie, Paris, 1985

[4] S. Belmehdi, “On semi-classical linear functionals of class $s=1$. Classification and integral representations”, Indag. Math., 3 (1992), 253–275 | DOI | MR | Zbl

[5] T. S. Chihara, “On Kernel polynomials and related systems”, Boll. Un. Mat. Ital., 19:3 (1964), 451–459 | MR | Zbl

[6] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978 | MR | Zbl

[7] W. Hahn, “Über differentialgleichungen für orthogonalpolynome [On differential equations for orthogonal polynomials]”, Monatsh. Math., 95:4 (1983), 269–274 | DOI | MR | Zbl

[8] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and Their Applications, 98, Cambridge University Press, Cambridge, 2005 | MR | Zbl

[9] M. E. H. Ismail, “More on electrostatic model for zeros of general orthogonal polynomials”, J. Nonlinear Funct. Anal. Opt., 21 (2000), 191–204 | DOI | MR | Zbl

[10] P. Maroni, “Le calcul des formes et les polynômes orthogonaux semi-classiques [Calculations of linear forms and semiclassical orthogonal polynomials]”, Orthogonal polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., 1329, eds. M. Alfaro et al., Springer-Verlag, Berlin, 1988, 279–290 | DOI | MR

[11] P. Maroni, “Sur la décomposition quadratique d'une suite de polynômes orthogonaux. I [On the quadratic decomposition of a sequence of orthogonal polynomials. I]”, Riv. Mat. Pura Appl., 6 (1990), 279–290 | MR

[12] P. Maroni, “Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques [An algebraic theory of orthogonal polynomials. Application to semi-classical orthogonal polynomials]”, Orthogonal polynomials and their applications (Erice, 1990), Comput. Appl. Math., 9, eds. C. Brezinski et al., Baltzer, Basel, 1991, 95–130 | MR | Zbl

[13] P. Maroni, “Variations around classical orthogonal polynomials. Connected problems”, J. Comput. Appl. Math., 48:1–2 (1993), 133–155 | DOI | MR | Zbl

[14] P. Maroni, “Fonctions eulériennes. Polynômes orthogonaux classiques”, Technique de l'ingénieur A, 154 (1994), 1–30

[15] P. Maroni, “An introduction to second degree forms”, Adv. Comput. Math., 3:1–2 (1995), 59–88 | DOI | MR | Zbl

[16] P. Maroni, M. Ihsen Tounsi, “The second-order self associate orthogonal polynomials”, J. Appl. Math., 2 (2004), 137–167 | DOI | MR | Zbl

[17] P. Maroni, M. Mejri, “The symmetric $D_\omega$-semi-classical orthogonal polynomials of class one”, Numer. Algorithms, 49:1 (2008), 251–282 | DOI | MR | Zbl

[18] S. Sghaier, “Some new results about a set of semi-classical polynomials os class $s=1$”, Int. Transf. Spec. Funct., 21:7 (2010), 529–539 | DOI | MR | Zbl

[19] J. Shohat, “A differential equation for orthogonal polynomials”, Duke Math. J., 5 (1939), 401–417 | DOI | MR | Zbl

[20] T. J. Stieltjes, “Recherches sur les fractions continues”, Ann. Fac. Sci. Toulouse, 8:4 (1894), 1–122 ; “Recherches sur les fractions continues [Suite et fin]”, 9:1 (1895), 5–47 | DOI | MR | Zbl | DOI | MR

[21] G. Szego, Orthogonal polynomials, American Mathematical Society Colloquium Publication, 23, 4th ed., American Mathematical Society, Providence, RI, 1975 | MR

[22] W. Van Assche, “The impact of Stieltjes work on continued fractions and orthogonal polynomials”, Collected Papers, ed. G. Van Dijk, Springer Verlag, Berlin, 1993, 5–37 | MR