Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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