Correlation hole of the spin-polarized electron gas, with exact small-wave-vector and high-density scaling

For a uniform electron gas of density n=n_+n_=3/4rₒ^3=kₒ^6/192 and spin polarization = (n_-n_) /n, we study the Fourier transform ₂ (k, rₒ, ) of the correlation hole, as well as the correlation energy ₂ (rₒ, ) =F₀^dk ₂/. In the high-density (rₒ0) limit, we find a simple scaling relation kₒ₂/g^2 (z, ), where z=k/gkₒ, g= (1+) ^2/3+ (1-) ^2/3/2, and f (z, 1) =f (z, 0). The function f (z, ) is only weakly dependent, and its small-z expansion -3z/^2+4 3 z^2/^2+. . . is also the exact small-wave-vector (k0) expansion for any rₒ or. Motivated by these considerations, and by a discussion of the large-wave-vector and low-density limits, we present two Pad\'e representations for ₂ at any k, rₒ, or, one within and one beyond the random-phase approximation (RPA). We also show that ₂^RPA obeys a generalization of Misawa's spin-scaling relation for ₂^RPA, and that the low-density (rₒ) limit of ₂^RPA is rₒ^-3/4.