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SIGMA 7 (2011), 107, 24 pages arXiv:0912.5447
https://doi.org/10.3842/SIGMA.2011.107
Properties of the Exceptional (Xl) Laguerre and Jacobi Polynomials
Choon-Lin Ho a, Satoru Odake b and Ryu Sasaki c
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
Received April 18, 2011, in final form November 19, 2011; Published online November 25, 2011
Abstract
We present various results on the properties of the
four infinite sets of the exceptional Xl polynomials
discovered recently by Odake and Sasaki [Phys. Lett. B 679 (2009), 414-417;
Phys. Lett. B 684 (2010), 173-176]. These Xl
polynomials are global solutions of second order Fuchsian
differential equations with l+3 regular singularities and
their confluent limits. We derive equivalent but much simpler
looking forms of the Xl 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 Xl polynomials.
Key words:
exceptional orthogonal polynomials; Gram-Schmidt process; Rodrigues formulas; generating functions.
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