  In the lights of current BICEP2 observations accompanied with the PLANCK
satellite results, it has been observed that the simple single field chaotic
inflationary models provide a good agreement with their spectral index n_s and
large tensor-to-scalar ratio r (0.15 <r <0.26). To explore the other simple
models, we consider the fractional-chaotic inflationary potentials of the form
V_0 phi^(a/b) where a and b are relatively prime. We show that such kind of
inflaton potentials can be realized elegantly in the supergravity framework
with generalized shift symmetry and a nature bound a/b < 4 for consistency.
Especially, for the number of e-folding from 50 to 60 and some a/b from 2 to 3,
our predictions are nicely within at least 1 $\sigma$ region in the r-n_s
plane. We also present a systematic investigation of such chaotic inflationary
models with fractional exponents to explore the possibilities for the
enhancement in the magnitude of running of spectral index (\alpha_{n_s}) beyond
the simplistic models.
