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Motivated by fractional quantum Hall effects, we introduce a universal spaceof statistics interpolating Bose-Einstein statistics and Fermi-Diracstatistics. We connect the interpolating statistics to umbral calculus and useit as a bridge to study the interpolation statistics by the principle maximumentropy by deformed entropy functions. On the one hand this connection makes itpossible to relate fractional quantum Hall effects to many differentmathematical objects, including formal group laws, complex bordism theory,complex genera, operads, counting trees, spectral curves in Eynard-Orantintopological recursions, etc. On the other hand, this also suggests to reexamineumbral calculus from the point of view of quantum mechanics and statisticalmechanics.

On Some Mathematics Related to the Interpolating Statistics