  Effects of short-range correlations on the Coulomb screening, the phonons,
and the pairing interactions are examined in electron-phonon systems. First, we
derive a model Hamiltonian of Coulomb interactions which includes both the
long-range part v_q and the short-range part U. It is found from the expression
of the dielectric function that the strong on-site correlations weaken the
Coulomb screening. Secondly, we examine the screened phonons and the
interaction mediated by phonons. In a consistent picture, we derive an
expression of the effective interaction which includes (1) the screened Coulomb
interactions, (2) the pairing interactions mediated by phonons, and (3) the
effective interactions mediated by spin and charge fluctuations. It is
rewritten in a form of a summation of (a) the effective interactions of the
pure Hubbard model without the long-range Coulomb interactions, and (b) the
phonon-mediated interactions plus screened Coulomb interactions with
corrections due to both U and v_q. Thirdly, we derive an effective Hamiltonian
analogous to the BCS Hamiltonian. Fourthly, for some typical values of
parameters, we obtain the ground state phase diagrams. It is found that
spin-triplet superconductivity mediated by phonons occurs when the short-range
electron correlations are sufficiently strong, and the Coulomb screening is
sufficiently weak. We estimate the orders of the transition temperatures when
the triplet superconductivity occurs. The obtained values are realistic for
existing candidates of the triplet superconductors as the order of the
magnitudes. The possible relevance of the phonon-mediated interactions to the
heavy fermion superconductor UPt_3 and the layered superconductors such
(TMTSF)_2X and Sr_2RuO_4 are briefly discussed.
