Abstract
Let Tc := D(x - c)((x - c)D + 2II) be a second-order linear differential operator, where c is an arbitrary complex number, \(D: = \frac{d}{{dx}}\) and II represents the identity on the linear space of polynomials with complex coefficients. The aim of this paper is to describe all of the Tc-classical orthogonal polynomials. Two canonical situations appear: the Laguerre \(\{L_n^{(2)}\}_{n\geq0}\) and the Jacobi \(\{P_n^{(\alpha-2,2)}\}_{n\geq0}\)
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References
F. Abdelkarim and P. Maroni, The Dw-classical orthogonal polynomials, Result. Math., 32 (1997), 1–28.
B. Aloui, F. Marcellán and R. Sfaxi, Classical orthogonal polynomials with respect to a lowering operator generalizing the Laguerre operator, Integral Transforms Spec. Funct., 24(8) (2013), 636–648.
B. Aloui, Hahn's problem with respect to a third-order differential operator, Transylv. J. Math. Mech., 6(2) (2014), 85–100.
B. Aloui, Characterization of Laguerre polynomials as orthogonal polynomials connected by the Laguerre degree raising operator, Ramanujan J., 45(2) (2018), 475–481.
W. Al-Salam, Characterization theorems for orthogonal polynomials, In: Orthogonal Polynomials: Theory and Practice, P. Nevai Editor, NATO ASI Series C, Vol. 294. Kluwer Acad. Publ., Dordrecht, (1990), 1–24.
W. Al-Salam and T. S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal., 3 (1972), 65–70.
I. Area, A. Godoy, A. Ronveaux and A. Zarzo, Classical symmetric orthogonal polynomials of a discrete variable, Integral Transforms Spec. Funct., 15 (2004), 1–12.
N. M. Atakishiyev, M. Rahman, and S. K. Suslov, On classical orthogonal polynomials, Constructive Approx., 11 (1995), 181–223.
Y. B. Cheikh and M. Gaied, Characterization of the Dunkl-classical symmetric orthogonal polynomials, Appl. Math. Comput., 187 (2007), 105–114.
S. Bochner, Über sturm-Liouvillesche polynomsysteme, Math Z., 29 (1929), 730–736.
A. Branquinho, J. Petronilho, and F. Marcellán, Classical orthogonal polynomials, a functional approach, Acta Appl. Math., 34 (1994), 283–303.
T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978.
E. Godoy, I. Area, A. Ronveaux, and A. Zarzo, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case, J. Comput. Appl. Math., 84 (1997), 257–275.
E. Grosswald, Bessel Polynomials, (Lecture Notes in Mathematics 698) Berlin: Springer, 1978.
V.W. Hahn, Über die Jacobischen polynome und zwei verwandte polynomklassen, Math. Z., 39 (1935), 634–638.
V.W. Hahn, Über Orthogonalpolynome, die q-Differenzengleichungen genügen, Math. Nach., 2 (1949), 4–34.
L. Khériji and P. Maroni, The Hq-classical orthogonal polynomialsm Acta. Appl. Math., 71 (2002), 49–115.
K. H. Kwon and G. J. Yoon, Generalized Hahn's theorem, J. Comput. Appl. Math., 116 (2000), 243–262.
N. N. Lebedev, Special functions and their applications, (Revised English Edition, Translated and Edited by Richard Silverman 1965), Dover Publications New York, 1972.
A. F. Loureiro and P. Maroni, Quadratic decomposition of Appell sequences, Expo. Math., 26 (2008), 177–186.
P. Maroni, Une théorie algébrique des polynômes orthogonaux Applications aux polynômes orthogonaux semi-classiques, In Orthogonal Polynomials and their Applications, C. Brezinski et al., Editors, IMACS Ann. Comput. Appl. Math., 9 (1991), 95–130.
P. Maroni, Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math., 48 (1993), 133–155.
P. Maroni, Fonctions Eulériennes, Polynômes Orthogonaux Classiques. Techniques de l'Ingénieur, Traité Généralités (Sciences Fondamentales) A 154 Paris. (1994), 1–30.
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer, Berlin, 1991.
A. Zarzo, I. Area, E. Godoy, and A. Ronveaux, Inversion problems for classical continuous and discrete orthogonal polynomials, J. Phys. A: Math. Gen., 30 (1997), L35–L40.
Acknowledgement
The authors would like to thank the referees for their corrections and many valuable suggestions. The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under Project Number No. (RGP-2019-5).
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Aloui, B., Chammam, W. Classical orthogonal polynomials via a second-order linear differential operators. Indian J Pure Appl Math 51, 689–703 (2020). https://doi.org/10.1007/s13226-020-0424-6
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DOI: https://doi.org/10.1007/s13226-020-0424-6
Key words
- Orthogonal polynomials
- quasi-definite linear functionals
- classical polynomials
- differential operators
- structure relations