  It was previously suggested that an odd-frequency pair amplitude exists in
the vicinity of boundaries in unconventional superconductors. We develop this
idea and quest for a novel superconducting order parameter with an
odd-frequency dependence. For this purpose, we focus on p-wave superconductors
and extend the quasi-classical theory to include the odd-frequency dependence
in the order parameter. Both of the frequency and spacial dependences of the
order parameter are determined self-consistently. Under a finite
electron-phonon interaction, it is found that an odd-frequency order parameter
is stabilized near the boundary and coexists with the even-frequency one. By
analyzing the induced odd-frequency pair amplitude in terms of the
superconducting quasi-particle wavefunction, it is found that the mid-gap bound
state generates the emergent odd-frequency order parameter.
