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This article is cited in 57 scientific papers (total in 57 papers)
Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials
Choon-Lin Hoa, Satoru Odakeb, Ryu Sasakic a Department of Physics, Tamkang University, Tamsui 251, Taiwan (R.O.C.)
b Department of Physics, Shinshu University, Matsumoto 390-8621, Japan
c Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract:
We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials
discovered recently by Odake and Sasaki [Phys. Lett. B 679 (2009), 414–417; Phys. Lett. B 684 (2010), 173–176]. These $X_{\ell}$ polynomials are global solutions of second order Fuchsian
differential equations with $\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape invariance, the forward and backward shift operations, invariant polynomial subspaces under the Fuchsian differential operators, the Gram–Schmidt
orthonormalisation procedure, three term recurrence relations and the generating functions for the $X_{\ell}$ polynomials.
Keywords:
exceptional orthogonal polynomials, Gram–Schmidt process, Rodrigues formulas, generating functions.
Received: April 18, 2011; in final form November 19, 2011; Published online November 25, 2011
Citation:
Choon-Lin Ho, Satoru Odake, Ryu Sasaki, “Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials”, SIGMA, 7 (2011), 107, 24 pp.
Linking options:
https://www.mathnet.ru/eng/sigma665 https://www.mathnet.ru/eng/sigma/v7/p107
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