Mathematical relations between impact factors and average number of citations

It is shown that both impact factors are equal if and only if the derivative of the impact-factor function is zero, and a simple classification of impact-Factor curves versus mean citation curves is established.

Abstract

Instead of the two-year impact factor as used in the Journal Citation Reports, there is much m favor of using x-year impact factors (x>0). These impact factors are studied as a function of x and compared with the average number of citations per paper to papers that appeared in the journal x years ago. It is shown that both are equal if and only if the derivative of the impact-factor function is zero. Based on this, a simple classification of impact-factor curves versus mean citation curves is established and examples are given. These results are also applied to recent practical data that were obtained by Rousseau.