Stability Method and a Depressed Cubic Solution beyond the Kaniadakis Quadratic?

Traditionally, the Tsallis and Kaniadakis distributions, p(x) unnormalized = (1+ (1-q)x) power 1/(1-q) and p(x) unnormalized= { sqrt(1+kkxx) + kx} power 1/k (with x= -ei/T) reduce to the Maxwell-Boltzmann solution for q=1 and k=0 and also for x<<1 (for any value of q,k). We ask, however: Is there any reason why a power law distribution should collapse to an MB solution for x<<1? In particular, in (1), we introduced a stability method which yields the MB, Tsallis and Kaniadakis distributions in a single unified approach. Here we try to see if one may have a depressed cubic solution (going beyond the quadratic form linked with the Kaniadakis case). We suggest that for x<<1, one does not obtain the MB distribution. Specifically, we suggest a stable distribution: p(ei) unnormalized = {cube root{ a1/2 + sqrt(a1*a1/4 +(a1*x) power 3/ 27)} + cube root{ a1/2 - sqrt(a1*a1/4 +(a1*x) power 3/ 27)} } power 1/2k For x= -ei/T>>1, this becomes a power law (dropping in x), but for x<<1 it does not become the Maxwell-Boltzmann distribution, but rather p(ei) unnormalized = (1- b1 x power 3) power 1/2k