The share of items of highly productive sources as a function of the size of the system

The research in this paper is based on the paper of D.W. Aksnes & G. Sivertsen: The effect of highly cited papers on national citation indicators, Scientometrics 59 (2) (2004), 213-224, where one states that "the few highly cited papers account for the highest share of the citations in the smallest fields". This, at first sight, evident property is examined in the theoretical models that exist in the literature. We first define exactly what we mean by "size of a field" (i.e. when is a field "smaller" or "larger" than another one). We show that there are two, non-equivalent possible definitions. Next we define exactly the possible property under study. This leads us again to two possible, non-equivalent formulations. Hence, in total, there are four different formulations to consider. We show, by giving counterexamples, that none of these four formulations are true in general. We also express conditions (in Lotkaian and Zipfian informetrics), under which the property of Aksnes and Sivertsen is true. All these results are not only valid in the papers-citations relationships but in any informetric source-item relationship. In this connection we present formulae describing the share of items of highly productive sources as a function of the parameters of the system (e.g. the size of the system).