Some Generalized Integrals Applied in Scientometrics and Related Evaluation

In this study, some examples show that Hirsch's index (h-index) usually has some limitation in evaluations. In order to provide more choices and diversities for scientometrics evaluation index, this study discusses four types of extended indexes. We firstly use a type of variation of Cognitive Integral to serve as a new generalized Hirsch's index, called Cognitive Hirsch's index, and analyze some of its mathematical properties. Secondly, by introducing Reference Function generalized Integral, we further generalize and extend Cognitive Hirsch's index. Next, this study also proposes to transform any citation sequence into a unique 0-1 function. Subsequently, based on different shapes, i.e., systems of some special 0-1 functions, this paper then proposes a novel index which is measured by counting the extent to which a citation sequence's 0-1 function can cover some given function systems. Lastly, still based on the 0-1 functions, we present a new index by using some predetermined distributions.