`ISRN Mathematical PhysicsVolume 2012 (2012), Article ID 920475, 27 pageshttp://dx.doi.org/10.5402/2012/920475`
Research Article

## Zeros of the Exceptional Laguerre and Jacobi Polynomials

1Department of Physics, Tamkang University, Tamsui 251, Taiwan
2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Received 12 April 2012; Accepted 4 July 2012

Academic Editors: G. Goldin and R. Schiappa

Copyright © 2012 Choon-Lin Ho and Ryu Sasaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. D. Gómez-Ullate, N. Kamran, and R. Milson, “An extended class of orthogonal polynomials defined by a Sturm-Liouville problem,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 352–367, 2009.
2. D. Gómez-Ullate David, N. Kamran, and R. Milson, “An extension of Bochner's problem: exceptional invariant subspaces,” Journal of Approximation Theory, vol. 162, no. 5, pp. 987–1006, 2010.
3. S. Odake and R. Sasaki, “Infinitely many shape invariant potentials and new orthogonal polynomials,” Physics Letters B, vol. 679, no. 4, pp. 414–417, 2009.
4. S. Odake and R. Sasaki, “Another set of infinitely many exceptional (${X}_{\ell }$) Laguerre polynomials,” Physics Letters B, vol. 684, no. 2-3, pp. 173–176, 2009.
5. S. Odake and R. Sasaki, “Infinitely many shape-invariant potentials and cubic identities of the Laguerre and Jacobi polynomials,” Journal of Mathematical Physics, vol. 51, no. 5, Article ID 053513, 9 pages, 2010.
6. C. Quesne, “Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry,” Journal of Physics A, vol. 41, no. 39, Article ID 392001, 6 pages, 2008.
7. C. Quesne, “Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics,” SIGMA, vol. 5, article 084, 24 pages, 2009.
8. B. Bagchi, C. Quesne, and R. Roychoudhury, “Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry,” Pramana—Journal of Physics, vol. 73, no. 2, pp. 337–347, 2009.
9. C. L. Ho, S. Odake, and R. Sasaki, “Properties of the exceptional (${X}_{\ell }$) Laguerre and Jacobi polynomials,” SIGMA, vol. 7, article 107, 24 pages, 2011.
10. D. Dutta and P. Roy, “Conditionally exactly solvable potentials and exceptional orthogonal polynomials,” Journal of Mathematical Physics, vol. 51, no. 4, Article ID 042101, 9 pages, 2010.
11. S. Bochner, “Über Sturm-Liouvillesche Polynomsysteme,” Mathematische Zeitschrift, vol. 29, no. 1, pp. 730–736, 1929.
12. S. Odake and R. Sasaki, “Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials,” Physics Letters B, vol. 682, no. 1, pp. 130–136, 2009.
13. S. Odake and R. Sasaki, “The Exceptional (${X}_{\ell }$) (q)-Racah Polynomials,” Progress of Theoretical Physics, vol. 125, no. 5, pp. 851–870, 2011.
14. D. Gómez-Ullate, N. Kamran, and R. Milson, “Exceptional orthogonal polynomials and the Darboux transformation,” Journal of Physics A, vol. 43, no. 43, Article ID 434016, 16 pages, 2010.
15. R. Sasaki, S. Tsujimoto, and A. Zhedanov, “Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations,” Journal of Physics A, vol. 43, no. 31, Article ID 315204, 20 pages, 2010.
16. C. L. Ho, “Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials,” Annals of Physics, vol. 326, no. 4, pp. 797–807, 2011.
17. G. Szego, Orthogonal Polynomials, vol. 23 of American Mathematical Society Colloquium Publications, American Mathematical Society, New York, NY, USA, 1939.
18. T. S. Chihara, An Introduction to Orthogonal Polynomials, vol. 13 of Mathematics and its Applications, Gordon and Breach Science, New York, NY, USA, 1978.
19. G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1999.
20. M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, vol. 98 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 2005.
21. Y. Grandati and A. Bérard, “Solvable rational extension of translationally shape invariantpotentials,” http://128.84.158.119/abs/0912.3061v2.
22. Y. Grandati, “Solvable rational extensions of the isotonic oscillator,” Annals of Physics, vol. 326, no. 8, pp. 2074–2090, 2011.
23. Y. Grandati, “Solvable rational extensions of the Morse and Kepler-Coulomb potentials,” Journal of Mathematical Physics, vol. 52, no. 10, Article ID 103505, 12 pages, 2011.