  We discuss some issues related to the computation of black hole entropy in
loop quantum gravity from the novel point of view provided by the recent
number-theoretical methods introduced by the authors and their collaborators.
In particular we give exact expressions, in the form of integral transforms,
for the black hole entropy in terms of the area. We do this by following
several approaches based both on our combinatorial techniques and also on
functional equations similar to those employed by Meissner in his pioneering
work on this subject. To put our results in perspective we compare them with
those of Meissner. We will show how our methods confirm some of his findings,
extend the validity of others, and correct some mistakes. At the end of the
paper we will discuss the delicate issue of the asymptotics of black hole
entropy.
