Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials

Details Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials

2009-12-30

arxiv.org/abs/0912.5447v4

SIGMA 7 (2011), 107, 24 pages

Physics

Mathematical Physics

Scientific paper

10.3842/SIGMA.2011.107

We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 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.

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