Properties of the Exceptional (X_l) Laguerre and Jacobi Polynomials
Abstract
We present various results on the properties of the four infinite sets of the exceptional X_{l} polynomials discovered recently by Odake and Sasaki [Phys. Lett. B 679 (2009), 414417; Phys. Lett. B 684 (2010), 173176]. These X_{l} 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 X_{l} 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 GramSchmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the X_{l} polynomials.
 Publication:

SIGMA
 Pub Date:
 November 2011
 DOI:
 10.3842/SIGMA.2011.107
 arXiv:
 arXiv:0912.5447
 Bibcode:
 2011SIGMA...7..107H
 Keywords:

 exceptional orthogonal polynomials;
 GramSchmidt process;
 Rodrigues formulas;
 generating functions;
 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Classical Analysis and ODEs;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 SIGMA 7 (2011), 107, 24 pages