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), 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 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 Gram-Schmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the X_l polynomials.

Keywords

• exceptional orthogonal polynomials
• Gram-Schmidt process
• Rodrigues formulas
• generating functions