A short note on the generalized logarithmic series distribution (Q1093985)

Let \(\beta\geq 1\) and \(0<\theta <\beta^{-1}\) and let P be the probability on \(\{\) 1,2,...\(\}\) charging x with \[ P_ x=(-\beta x\quad \ln (1-\theta))^{-1}\left( \begin{matrix} \beta x\\ x\end{matrix} \right)\theta^ x(1-\theta)^{\beta x-x}, \] where \(\left( \begin{matrix} a\\ b\end{matrix} \right)=\Gamma (a+1)/(\Gamma (b+1)\Gamma (a-b+1))\). The author proves by direct computation that P is unimodal with mode 1 but is not strongly unimodal (s.u.); s.u. meaning \(P^ 2_ x/(P_{x- 1}P_{x+1})\geq 1\) for all \(x\geq 2\) or, equivalently, as is known, that the convolutions of P with all unimodal distributions are unimodal. Indications concerning the importance of the result are given.