In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, nd logarithmically complete monotonicity of these ...

We define the incomplete gamma function ratio for positive N by \begin{align*}\tag{1}P(N, b, X) &= \int^X_0 t^{N-1}e^{-bt} dt/ \int^\infty_0 t^{N-1}e^{-bt} dt ...