  Motivated by fractional quantum Hall effects, we introduce a universal space
of statistics interpolating Bose-Einstein statistics and Fermi-Dirac
statistics. We connect the interpolating statistics to umbral calculus and use
it as a bridge to study the interpolation statistics by the principle maximum
entropy by deformed entropy functions. On the one hand this connection makes it
possible to relate fractional quantum Hall effects to many different
mathematical objects, including formal group laws, complex bordism theory,
complex genera, operads, counting trees, spectral curves in Eynard-Orantin
topological recursions, etc. On the other hand, this also suggests to reexamine
umbral calculus from the point of view of quantum mechanics and statistical
mechanics.
