  A previous study of exactly solvable rationally-extended radial oscillator
potentials and corresponding Laguerre exceptional orthogonal polynomials
carried out in second-order supersymmetric quantum mechanics is extended to
$k$th-order one. The polynomial appearing in the potential denominator and its
degree are determined. The first-order differential relations allowing one to
obtain the associated exceptional orthogonal polynomials from those arising in
a ($k-1$)th-order analysis are established. Some nontrivial identities
connecting products of Laguerre polynomials are derived from shape invariance.
