  The purpose of this communication is to point out the connection between a 1D
quantum Hamiltonian involving the fourth Painleve transcendent P$_{\rm IV}$,
obtained in the context of second-order supersymmetric quantum mechanics and
third-order ladder operators, with a hierarchy of families of quantum systems
called $k$-step rational extensions of the harmonic oscillator and related with
multi-indexed $X_{m_{1},m_{2},...,m_{k}}$ Hermite exceptionnal orthogonal
polynomials of type III. The connection between these exactly solvable models
is established at the level of the equivalence of the Hamiltonians using
rational solutions of the fourth Painleve equation in terms of generalized
Hermite and Okamoto polynomials. We also relate the different ladder operators
obtained by various combinations of supersymmetric constructions involving
Darboux-Crum and Krein-Adler supercharges, their zero modes and the
corresponding energies. These results will demonstrate and clarify the relation
observed for a particular case in previous papers.
