  Many discussions have enlarged the literature in Bibliometrics since the
Hirsh proposal, the so called $h$-index. Ranking papers according to their
citations, this index quantifies a researcher only by its greatest possible
number of papers that are cited at least $h$ times. A closed formula for
$h$-index distribution that can be applied for distinct databases is not yet
known. In fact, to obtain such distribution, the knowledge of citation
distribution of the authors and its specificities are required. Instead of
dealing with researchers randomly chosen, here we address different groups
based on distinct databases. The first group is composed by physicists and
biologists, with data extracted from Institute of Scientific Information (ISI).
The second group composed by computer scientists, which data were extracted
from Google-Scholar system. In this paper, we obtain a general formula for the
$h$-index probability density function (pdf) for groups of authors by using
generalized exponentials in the context of escort probability. Our analysis
includes the use of several statistical methods to estimate the necessary
parameters. Also an exhaustive comparison among the possible candidate
distributions are used to describe the way the citations are distributed among
authors. The $h$-index pdf should be used to classify groups of researchers
from a quantitative point of view, which is meaningfully interesting to
eliminate obscure qualitative methods.
