Classification of Exceptional Jacobi Polynomials

Abstract

We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification containssix degeneracy classes according to whether 𝛼, 𝛽 or 𝛼 ± 𝛽 assume integer values. Exceptional Jacobi operatorsare in one-to-one correspondence with spectral diagrams, a combinatorial object that describes the quasi-rationaleigenfunctions of the operator and their asymptotic behavior at the endpoints of (−1, 1). With a convenient indexingscheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptionaloperators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class 𝛼, 𝛽 ∈ℕ0 , there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classificationresult is achieved by a careful description of all possible rational Darboux transformations that can be performed onexceptional Jacobi operators.