The stuttering generalized Waring distribution (Q1083106)

The generalized Waring distribution is defined by the probability function \[ (1)\quad P(X=x)=\frac{c_{(m)}}{(a+c)_{(m)}}\frac{a_{(x)}m_{(x)}}{(a+m+c)_ {(x)}}\frac{1}{x!},\quad x=0,1,...,\quad a,m,c>0, \] where \(^{\alpha}(\beta)=\Gamma (\alpha +\beta)/\Gamma (\alpha)\), \(\alpha >0\), \(\beta\in {\mathbb{R}}\). The authors introduce the stuttering generalized Waring distribution as an extension of (1) and show that it arises through two urn genesis schemes. The generating function and the moments are derived and some potential applications are discussed in areas such as bibliographic research, linguistics, ecology and inventory control.