Classification of Exceptional Jacobi Polynomials

ABSTRACT We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether or assume integer values. Exceptional Jacobi operators are in one‐to‐one correspondence with spectral diagrams, a combinatorial object that describes the quasi‐rational eigenfunctions of the operator and their asymptotic behavior at the endpoints of . With a convenient indexing scheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptional operators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class , there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classification result is achieved by a careful description of all possible rational Darboux transformations that can be performed on exceptional Jacobi operators.