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A previous study of exactly solvable rationally-extended radial oscillatorpotentials and corresponding Laguerre exceptional orthogonal polynomialscarried out in second-order supersymmetric quantum mechanics is extended to$k$th-order one. The polynomial appearing in the potential denominator and itsdegree are determined. The first-order differential relations allowing one toobtain the associated exceptional orthogonal polynomials from those arising ina ($k-1$)th-order analysis are established. Some nontrivial identitiesconnecting products of Laguerre polynomials are derived from shape invariance.

Rationally-extended radial oscillators and Laguerre exceptional orthogonal polynomials in kth-order SUSYQM