  Motivated by string theory activities, we investigate inflationary models and
the swampland criteria in the context of a stringy rescaled gravity.
  Inspired by differential equations associated with special functions, we
develop an algorithm to derive new scalar potential functions with
hypergeometric behaviors from string theory correction terms. Among others, we
obtain a family of models indexed by a couple $(m,n)$, where $m$ and $n$ are
natural numbers constrained by hypergeometric behaviors and certain physical
requirements. Using the falsification scenario, we confront the derived models
with the Planck observational data for such a stringy rescaled gravity. Then,
we approach the associated swampland conjectures. For certain models of
phenomenological interest, we find that the swampland criteria are satisfied
for small values of the slow-roll parameters in such a modified gravity.
