Properties of the Exceptional () Laguerre and Jacobi Polynomials
arXiv:0912.5447 · doi:10.3842/SIGMA.2011.107
Abstract
We present various results on the properties of the four infinite sets of the exceptional 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 polynomials are global solutions of second order Fuchsian differential equations with regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the 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 polynomials.
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- Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
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- A class of exactly solvable rationally extended non-central potentials in Two and Three Dimensions
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- Exactly and quasi-exactly solvable `discrete' quantum mechanics
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