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.

To appear in Physica A (8 pages, 6 figures and 2 tables)