@MarkL.Stone and Iosif , can I ask you two to please take a look at my question over here? math.stackexchange.com/questions/4757229/… I discovered this question while hunting for an answer to my own and I'm hopeful one of you two might know how to answer my question

When $a = 1$, how does $\mathbb{E}[log (1 - x)]$ become that difference of polygamma functions?

Thank you! I have a small related question: is there some way to interpret $\frac{t^{r-1}}{\Gamma(r)}$ as a probability distribution over $t$?