Geodesic
Characterization of polynomials via a raising operator
Problemy analiza, Tome 13 (2024) no. 1, pp. 71-81.

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This paper investigates a first-order linear differential operator $\mathcal{J}_\xi$, where $\xi=(\xi_1, \xi_2) \in \mathbb{C}^2\setminus{(0, 0)}$, and $D:=\frac{d}{dx}$. The operator is defined as $\mathcal{J}_{\xi}:=x(xD+\mathbb{I})+\xi_1\mathbb{I}+\xi_2 D$, with $\mathbb{I}$ representing the identity on the space of polynomials with complex coefficients. The focus is on exploring the $\mathcal{J}_\xi$-classical orthogonal polynomials and analyzing properties of the resulting sequences. This work contributes to the understanding of these polynomials and their characteristics.
Keywords: СЃlassical polynomials, second-order differential equation, raising operator.
Mots-clés : orthogonal polynomials 
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J. Souissi. Characterization of polynomials via a raising operator. Problemy analiza, Tome 13 (2024) no. 1, pp. 71-81. http://geodesic.mathdoc.fr/item/PA_2024_13_1_a4/

[1] Aloui B., “Chebyshev polynomials of the second kind via raising operator preserving the orthogonality”, Period. Math. Hung., 76 (2018), 126–132 | DOI | MR | Zbl

[2] Aloui B., Khériji L., “A note on the Bessel form of parameter 3/2”, Transylv. J. Math. Mech., 2019, no. 11, 09–13

[3] Aloui B., Souissi J., “Hahn's problem with respect to some perturbations of the raising operator $X-c$”, Ural. Math. J., 6:2 (2020), 15–24 | DOI | MR | Zbl

[4] Atia M.J., Alaya J., “Some classical polynomials seen from another side”, Period. Math. Hung., 38:1-2 (1999), 1–13 | DOI | MR | Zbl

[5] Böchner S., “Über Sturm-Liouvillesche Polynomsysteme”, Z. Math., 29 (1929), 730–736 | DOI | MR

[6] Chaggara H., “Operational rules and a generalized Hermite polynomials”, J. Math. Anal. Appl., 332 (2007), 11–21 | DOI | MR | Zbl

[7] Chihara T. S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978 | MR

[8] Dattoli G., Ricci P. E., “Laguerre-type exponentials, and the relevant L-circular and L-hyperbolic functions”, Georgian Math. J., 2003, no. 10, 481–494 | DOI | MR | Zbl

[9] Hahn W., “Über die jacobischen polynome und zwei verwandte polynomklassen”, Math. Z., 39 (1935), 634–638 | DOI | MR | Zbl

[10] Koornwinder T. H., “Lowering and raising operators for some special orthogonal polynomials”, Jack, Hall-Littlewood and Macdonald Polynomials, Contemporary Mathematics, 417, 2006 | DOI | MR | Zbl

[11] Maroni P., “Une théorie algébrique des polynô mes orthogonaux Applications aux polynômes orthogonaux semi-classiques”, Orthogonal Polynomials and their Applications, IMACS Ann. Comput. Appl. Math., 9, eds. C. Brezinski et al., 1991, 95–130 | MR | Zbl

[12] Maroni P., “Variations autour des polynômes orthogonaux classiques”, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), 209–212 | MR | Zbl

[13] Maroni P., “Fonctions Eulériennes, Polynômes Orthogonaux Classiques”, Techniques de l'Ingénieur, Traité Gén éralités (Sciences Fondamentales), A 154, Paris, 1994, 1–30

[14] Sonine N. J., “On the approximate computation of definite integrals and on the entire functions occurring there”, Warsch. Univ. Izv., 1887, no. 18, 1–76

[15] Srivastava H. M., Ben Cheikh Y., “Orthogonality of some polynomial sets via quasi-monomiality”, Appl. Math. Comput., 141 (2003), 415–425 | DOI | MR | Zbl