A didactic historical review of the distributions using the Bessel function: some extensions with unification

Based on a didactic historical review of the distributions using the Bessel function, some new extensions with unification are shown. Their probability density functions (pdf’s) are given by several integral representations of the Bessel function, whose distributions are called the basic Bessel (BB) ones, including the known Halphen type A. Power transformations of the BB (PBB) distributions are provided with their moments. The differential equations for the PBB distributions are shown and used to have their modes. A unified expression for unimodal extended distributions is also given with the differential equation.