Prabhakar Functions of Le Roy Type: Inequalities and Asymptotic Formulae - Jordanka Paneva-Konovska (Mathematics, 2023)
In this paper, the four-index generalization of the classical Le Roy function is considered on a wider set of parameters and its order and type are given. Letting one of the parameters take non-negative integer values, a family of functions with such a type of index is constructed. The behaviour of these functions is studied in the complex plane <semantics>C</semantics> and in different domains thereof. First, several inequalities are obtained in <semantics>C</semantics>, and then they are modified on its compact subsets as well. Moreover, an asymptotic formula is proved for ‘large’ values of the indices of these functions. Additionally, the multi-index analogue of the abovementioned four-index Le Roy type function is considered and its basic properties are obtained. Finally, several special cases of the two functions under consideration are discussed.