  Mathematical notation, i.e., the writing system used to communicate concepts
in mathematics, encodes valuable information for a variety of information
search and retrieval systems. Yet, mathematical notations remain mostly
unutilized by today's systems. In this paper, we present the first in-depth
study on the distributions of mathematical notation in two large scientific
corpora: the open access arXiv (2.5B mathematical objects) and the mathematical
reviewing service for pure and applied mathematics zbMATH (61M mathematical
objects). Our study lays a foundation for future research projects on
mathematical information retrieval for large scientific corpora. Further, we
demonstrate the relevance of our results to a variety of use-cases. For
example, to assist semantic extraction systems, to improve scientific search
engines, and to facilitate specialized math recommendation systems. The
contributions of our presented research are as follows: (1) we present the
first distributional analysis of mathematical formulae on arXiv and zbMATH; (2)
we retrieve relevant mathematical objects for given textual search queries
(e.g., linking $P_{n}^{(\alpha, \beta)}\!\left(x\right)$ with `Jacobi
polynomial'); (3) we extend zbMATH's search engine by providing relevant
mathematical formulae; and (4) we exemplify the applicability of the results by
presenting auto-completion for math inputs as the first contribution to math
recommendation systems. To expedite future research projects, we have made
available our source code and data.
