Properties of the Exceptional ( X l ) Laguerre and Jacobi Polynomials

[摘要] 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 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 Gram-Schmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the X l polynomials.

[关键词] exceptional orthogonal polynomials;Gram–Schmidt process;Rodrigues formulas;generating functions