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 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.
- Publication:
-
SIGMA
- Pub Date:
- November 2011
- DOI:
- 10.3842/SIGMA.2011.107
- arXiv:
- arXiv:0912.5447
- Bibcode:
- 2011SIGMA...7..107H
- Keywords:
-
- exceptional orthogonal polynomials;
- Gram-Schmidt process;
- Rodrigues formulas;
- generating functions;
- Mathematical Physics;
- High Energy Physics - Theory;
- Mathematics - Classical Analysis and ODEs;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- SIGMA 7 (2011), 107, 24 pages