I'm afraid my math in this area is a bit deficient, but what I am trying to do is see if there's a way to simplify the function: but I never really learned how to manipulate factorials algebraically, ...

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 ...

We prove among others results that the harmonic mean of Γq(ₓ) and Γq(1/ₓ) is greater than or equal to 1 for arbitrary x > 0, and q ∈ J where J is a subset of [0, +∞). Also, we prove that there is a ...