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_↓=3/4{π}rₛ³={π}kₛ⁶/192 and spin polarization {ζ}=(n_↑-n_↓)/n, we study the Fourier transform {ρ}c(k,rₛ,{ζ}) of the correlation hole, as well as the correlation energy εc(rₛ,{ζ})=F₀^∞dk {ρ}c/{π}. In the high-density (rₛ{→}0) limit, we find a simple scaling relation kₛ{ρ}c/{π}g²{→}f(z,{ζ}), where z=k/gkₛ, g=[(1+{ζ})2/3+(1-{ζ})2/3]/2, and f(z,1)=f(z,0). The function f(z,{ζ}) is only weakly {ζ} dependent, and its small-z expansion -3z/π²+4 {}3 z²/π²+... is also the exact small-wave-vector (k{→}0) expansion for any rₛ or {ζ}. Motivated by these considerations, and by a discussion of the large-wave-vector and low-density limits, we present two Pad\'e representations for {ρ}c at any k, rₛ, or {ζ}, one within and one beyond the random-phase approximation (RPA). We also show that {ρ}{} cRPA obeys a generalization of Misawa's spin-scaling relation for εcRPA, and that the low-density (rₛ{→}{∞}) limit of εcRPA is {~}rₛ^-3/4.