connected with a high frequency cut-off of the fluctuations. So all e-e processes we consider may be called Excitonically induced pairing. But all empirical information on high T$_{c}$ induced through e-e processes indicates that the ’excitons’ must be particle-hole fluctuations of the same one-particle excitations that pair up and such that they engender singularities in the single-particle spectra so that the distinction between particle-hole fluctuations of the fermions and the ’excitons’ is lost. For a general spectral function, the Eliashberg equations require a numerical solution. For the limited purposes of this paper, it is enough to use the McMillan simplified solution [25], where by the transition temperature in the s-wave channel is given for the e-ph problem by with where N (0) is the density of states of un-renormalized electrons at the fermi-surface, < ω > is an average over the phonon frequencies, κ is a (suitably defined) average over the lattice stiffness, κ = M < ω 2 > , and < I 2 > is the scattering averaged over the spectral function of the phonons: where a spin-trace (in the singlet channel) of g ν $_{α,β}$( k , k + q ) g ν $_{γ,δ}$( − k , − k − q ) has been taken to obtain | g ν k , q | $^{2}$. It is important to note that the λ of Eq.(23) also gives the mass renormalization in the normal state due to phonons so that the renormalization of the specific heat coefficient due to interaction with phonons is 27 ( m $^{∗}$/m )$_{ph}$ = (1 + λ e − $^{ph}$) . (25) < I 2 > = ∫ dS$_{k}$ ∫ dS$_{k}$ ′ ∑ ν | g ν k , q | $^{2}$v − 1 k v − 1 k ′ ∫ dS$_{k}$ ∫ dS$_{k}$ ′ v − 1 k v − 1 k ′ (24) λ e − ph ≡ N (0) < I 2 > κ , (23) T$_{c}$ ≈ c exp( − (1 + λ e − $^{ph}$) /λ e − $^{ph}$) , (22) B$_{α,β,γ,δ}$ ( q , ν, Ω) ≡ ∫ ∞ −∞ dω F $_{α,β,γ,δ}$( q , ν, ω ) Ω − ω , (21) A. McMillan’s Expression for the Coupling constant III. ELECTRON-PHONON INTERACTION PROMOTED T$_{c}$