Recurrence Relations and General Solution of the Exceptional Hermite Equation

Exceptional orthogonal Hermite polynomials have been linked to the k-step extension of the harmonic oscillator. The exceptional polynomials allow the existence of different supercharges in the Darboux–Crum and Krein–Adler constructions. They also allow the existence of different types of ladder relations and their associated recurrence relations. The existence of such relations is a unique property of these polynomials. Those relations have been used to construct 2D models which are superintegrable and display an interesting spectrum, degeneracies and finite-dimensional unitary representations. In previous works, only the physical or polynomial part of the spectrum was discussed. It is known that the general solutions are associated with other types of recurrence/ladder relations. We discuss in detail the case of the exceptional Hermite polynomials \(X_2^{(1)}\) and explicitly present new chains obtained by acting with different types of ladder operators. We exploit a recent result by one of the authors (Chalifour and Grundland in Ann Henri Poincaré 21:3341, 2020), where the general analytic solution was constructed and connected with the non-degenerate confluent Heun equation. The analogue Rodrigues formulas for the general solution are constructed. The set of finite states from which the other states can be obtained algebraically is not unique, but the vanishing arrow and diagonal arrow from the diagram of the 2-chain representations can be used to obtain minimal sets. These Rodrigues formulas are then exploited, not only to construct the states, polynomial and non-polynomial, in a purely algebraic way, but also to obtain coefficients from the action of ladder operators in an algebraic manner based on further commutation relations between monomials in the generators.