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The type III Hermite $X_m$ exceptional orthogonal polynomial family isgeneralized to a double-indexed one $X_{m_1,m_2}$ (with $m_1$ even and $m_2$odd such that $m_2 > m_1$) and the corresponding rational extensions of theharmonic oscillator are constructed by using second-order supersymmetricquantum mechanics. The new polynomials are proved to be expressible in terms ofmixed products of Hermite and pseudo-Hermite ones, while some of the associatedpotentials are linked with rational solutions of the Painlev\'e IV equation. Anovel set of ladder operators for the extended oscillators is also built andshown to satisfy a polynomial Heisenberg algebra of order $m_2-m_1+1$, whichmay alternatively be interpreted in terms of a special type of$(m_2-m_1+2)$th-order shape invariance property.

Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators