Exceptional Charlier and Hermite orthogonal polynomials. (English) Zbl 1298.33016
J. Approx. Theory 182, 29-58 (2014); corrigendum ibid. 253, Article ID 105349, 5 p. (2020).
The so-called exceptional orthogonal polynomials have been firstly considered by Gómez-Ullate, Kamran, and Milson in a series of papers and its study has been know an increasing interest after the works of Quesne where the connection with exactly solvable models in Quantum Mechanics was stablished. The study of the exceptional orthogonal polynomials was concentrated on the so-called continuous case (extensions of the classical orthogonal polynomials of Hermite, Laguerre and Jacobi) and some few examples related to the basic hypergeometric polynomials. In this study remained a gap concerning the study of the discrete case, i.e., the extension of the Meixner, Charlier, Kravchuk and Hahn polynomials. In the present paper the author filled that gap, showing that it is possible to construct the exceptional discrete polynomials, but not only this. He discovered a very interesting connection between the Casorati determinants of Charlier polynomials, the duality property of orthogonal polynomials and the Krall discrete orthogonal polynomials. Namely, the exceptional polynomials are the duals of the Krall discrete orthogonal polynomials (very recently studied by the same author). Finally, by taking appropriate limits he obtained several results on the exceptional Hermite polynomials.
Reviewer: Renato Alvarez-Nodarse (Sevilla)
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 33C47 | Other special orthogonal polynomials and functions |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
classical orthogonal polynomials; exceptional orthogonal polynomial; Charlier polynomials; Hermite polynomialsReferences:
| [1] | Adler, V. E., A modification of Crum’s method, Theoret. Math. Phys., 101, 1381-1386 (1994) · Zbl 0857.34069 |
| [2] | Akhiezer, N. I., The Classical Moment Problem (1965), Oliver & Boyd: Oliver & Boyd Edinburgh · Zbl 0135.33803 |
| [3] | Atkinson, F. V., Discrete and Continuous Boundary Problems (1964), Academic Press: Academic Press NY · Zbl 0117.05806 |
| [4] | Bochner, S., Über Sturm-Liouvillesche polynomsysteme, Math. Z., 29, 730-736 (1929) · JFM 55.0260.01 |
| [5] | Cariñena, J. F.; Perelomov, A. M.; Rañada, M. F.; Santander, M., A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator, J. Phys. A, 41, 085301 (2008) · Zbl 1138.81380 |
| [6] | Chihara, T., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach Science Publishers · Zbl 0389.33008 |
| [7] | Christoffel, E. B., Über die Gaussische Quadratur und eine Verallgemeinerung derselben, J. Reine Angew. Math., 55, 61-82 (1858) · ERAM 055.1450cj |
| [9] | Dubov, S. Y.; Eleonskii, V. M.; Kulagin, N. E., Equidistant spectra of anharmonic oscillators, Sov. Phys. JETP. Sov. Phys. JETP, Chaos, 4, 47-53 (1994) · Zbl 1055.81529 |
| [10] | Durán, A. J., Orthogonal polynomials satisfying higher order difference equations, Constr. Approx., 36, 459-486 (2012) · Zbl 1264.42006 |
| [11] | Durán, A. J., Using \(D\)-operators to construct orthogonal polynomials satisfying higher order difference or differential equations, J. Approx. Theory, 174, 10-53 (2013) · Zbl 1286.34114 |
| [12] | Durán, A. J., Symmetries for Casorati determinants of classical discrete orthogonal polynomials, Proc. Amer. Math. Soc., 142, 915-930 (2014) · Zbl 1284.42079 |
| [13] | Durán, A. J., Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities, J. Combin. Theory Ser. A., 124, 57-96 (2014) · Zbl 1292.33013 |
| [14] | Durán, A. J.; de la Iglesia, M. D., Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Constr. Approx. (2014), to appear. arXiv:1307.1326 |
| [15] | Dutta, D.; Roy, P., Conditionally exactly solvable potentials and exceptional orthogonal polynomials, J. Math. Phys., 51, 042101 (2010) · Zbl 1310.81076 |
| [16] | Gantmacher, F. R., The Theory of Matrices (1960), Chelsea Publishing Company: Chelsea Publishing Company New York · Zbl 0085.01001 |
| [17] | Gómez-Ullate, D.; Kamran, N.; Milson, R., An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl., 359, 352-367 (2009) · Zbl 1183.34033 |
| [18] | Gómez-Ullate, D.; Kamran, N.; Milson, R., An extension of Bochner’s problem: exceptional invariant subspaces, J. Approx. Theory, 162, 987-1006 (2010) · Zbl 1214.34079 |
| [19] | Gómez-Ullate, D.; Kamran, N.; Milson, R., Exceptional orthogonal polynomials and the Darboux transformation, J. Phys. A, 43, 434016 (2010) · Zbl 1202.81036 |
| [20] | Gómez-Ullate, D.; Kamran, N.; Milson, R., On orthogonal polynomials spanning a non-standard flag, Contemp. Math., 563, 51-71 (2012) · Zbl 1247.33016 |
| [21] | Gómez-Ullate, D.; Kamran, N.; Milson, R., A conjecture on exceptional orthogonal polynomials, Found. Comput. Math., 13, 615-666 (2013) · Zbl 1282.33019 |
| [22] | Gómez-Ullate, D.; Grandati, Y.; Milson, R., Rational extensions of the quantum Harmonic oscillator and exceptional Hermite polynomials, J. Phys. A, 47, 015203 (2014) · Zbl 1283.81059 |
| [23] | Karlin, S.; McGregor, J. L., Coincidence properties of birth and death processes, Pacific J. Math., 9, 1109-1140 (1959) · Zbl 0097.34102 |
| [24] | Karlin, S.; Szegő, G., On certain determinants whose elements are orthogonal polynomials, J. Anal. Math., 8, 1-157 (1961) · Zbl 0116.27901 |
| [25] | Krein, M. G., A continual analogue of a Christoffel formula from the theory of orthogonal polynomials, Dokl. Akad. Nauk. SSSR, 113, 970-973 (1957) · Zbl 0151.11202 |
| [26] | Koekoek, R.; Lesky, P. A.; Swarttouw, L. F., Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues (2008), Springer Verlag: Springer Verlag Berlin |
| [27] | Leonard, D., Orthogonal polynomials, duality, and association schemes, SIAM J. Math. Anal., 13, 656-663 (1982) · Zbl 0495.33006 |
| [28] | Midya, B.; Roy, B., Exceptional orthogonal polynomials and exactly solvable potentials in position dependent mass Schrödinger Hamiltonians, Phys. Lett. A, 373, 4117-4122 (2009) · Zbl 1234.81072 |
| [29] | Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable (1991), Springer Verlag: Springer Verlag Berlin · Zbl 0743.33001 |
| [30] | Odake, S.; Sasaki, R., Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B, 679, 414-417 (2009) |
| [31] | Odake, S.; Sasaki, R., Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials, Phys. Lett. B, 682, 130-136 (2009) |
| [32] | Odake, S.; Sasaki, R., The exceptional \((X_\ell)(q)\)-Racah polynomials, Prog. Theor. Phys., 125, 851-870 (2011) · Zbl 1223.33035 |
| [33] | Odake, S.; Sasaki, R., Dual Christoffel transformations, Prog. Theor. Phys., 126, 1-34 (2011) · Zbl 1230.81031 |
| [34] | Odake, S.; Sasaki, R., Multi-indexed \(q\)-Racah polynomials, J. Phys. A, 45 (2012), 385201 (21pp) · Zbl 1252.81053 |
| [35] | Quesne, C., Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A, 41, 392001-392007 (2008) · Zbl 1192.81174 |
| [36] | Sasaki, R.; Tsujimoto, S.; Zhedanov, A., Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations, J. Phys. A: Math. Gen., 43, 315204 (2010) · Zbl 1200.33013 |
| [37] | Szegö, G., Orthogonal Polynomials (1959), American Mathematical Society: American Mathematical Society Providence, RI · JFM 61.0386.03 |
| [38] | Tanaka, T., N-fold Supersymmetry and quasi-solvability associated with X2-Laguerre polynomials, J. Math. Phys., 51, 032101 (2010) · Zbl 1309.81102 |
| [39] | Yermolayeva, O.; Zhedanov, A., Spectral transformations and generalized Pollaczek polynomials, Methods Appl. Anal., 6, 261-280 (1999) · Zbl 0956.33006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
