Characterisation of the $\chi$-index and the $rec$-index

Axiomatic characterisation of a bibliometric index provides insight into the properties that the index satisfies and facilitates the comparison of different indices. A geometric generalisation of the $h$-index, called the $\chi$-index, has recently been proposed to address some of the problems with the $h$-index, in particular, the fact that it is not scale invariant, i.e., multiplying the number of citations of each publication by a positive constant may change the relative ranking of two researchers. While the square of the $h$-index is the area of the largest square under the citation curve of a researcher, the square of the $\chi$-index, which we call the $rec$-index (or {\em rectangle}-index), is the area of the largest rectangle under the citation curve. Our main contribution here is to provide a characterisation of the $rec$-index via three properties: {\em monotonicity}, {\em uniform citation} and {\em uniform equivalence}. Monotonicity is a natural property that we would expect any bibliometric index to satisfy, while the other two properties constrain the value of the $rec$-index to be the area of the largest rectangle under the citation curve. The $rec$-index also allows us to distinguish between {\em influential} researchers who have relatively few, but highly-cited, publications and {\em prolific} researchers who have many, but less-cited, publications.