The beta family includes the beta of the first and second kindStuart, A. and Ord, J.K. (1987): Kendall's Advanced Theory of Statistics, New York: Oxford University Press. (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting c = 0, b = 1 yields the standard two-parameter beta distribution.
