  A general class of functionally-fitted explicit pseudo two-step
Runge-Kutta-Nystr\"{o}m (FEPTRKN) methods for solving second-order initial
value problems has been studied. These methods can be considered generalized
explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (EPTRKN) methods. We proved
that an $s$-stage FEPTRKN method has step order $p = s$ and stage order $r = s$
for any set of distinct collocation parameters $(c_i)_{i=1}^s$.
Supperconvergence for the accuracy orders of these methods can be obtained if
the collocation parameters $(c_i)_{i=1}^s$ satisfy some orthogonality
conditions. We proved that an $s$-stage FEPTRKN method can attain accuracy
order $p = s + 3$. Numerical experiments have shown that the new FEPTRKN
methods work better than do EPTRKN methods on problems whose solutions can be
well approximated by the functions in bases on which these FEPTRKN methods are
developed.
