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Exceptional orthogonal Hermite and Laguerre polynomials have been linked tothe k-step extension of harmonic and singular oscillators. The exceptionalpolynomials allow the existence of different supercharges from the Darboux-Crumand Krein-Adler constructions of supersymmetric quantum mechanics. They alsoallow the existence of different types of ladder relations and their associatedrecurrence relations. The existence of such relations is a unique property ofthese polynomials. Those relations have been used to construct 2D models whichare superintegrable, and display an interesting spectrum, degeneracies andfinite-dimensional unitary representations. In previous works, only thephysical or polynomial part of the spectrum is discussed. It is known that thegeneral solutions are associated with other types of recurrence/ladderrelations. We plan to discuss in detail the case of the exceptional Hermitepolynomials $X_2^{(1)}$ and to explicitly present new chains obtained by actingwith different types of ladder operators. We will exploit a recent result byone of the authors [32], where the general analytic solution was constructedand connected with the non-degenerate confluent Heun equation. The analogueRodrigues formulas for the general solution are constructed. The set of finitestates from which the other states can be obtained algebraically is not uniquebut the vanishing arrow and diagonal arrow from the diagram of the 2-chainrepresentations can be used to obtain minimal sets. These Rodrigues formulasare then exploited, not only to construct the states, polynomial andnon-polynomial, in a purely algebraic way, but also to obtain coefficients fromthe action of ladder operators also in an algebraic manner based on furthercommutation relations between monomials of the generators.

Recurrence relations and general solution of the exceptional Hermite equation