On the Density arising from the Domain of Attraction between Sum and Supremum: the $\alpha$-Sun operator

Abstract

We explore the analytic properties of the density function $ h(x;\gamma,\alpha) $, $ x \in (0,\infty) $, $ \gamma > 0 $, $ 0 < \alpha < 1 $ which arises from the domain of attraction problem for a statistic interpolating between the supremum and sum of random variables. The parameter $ \alpha $ controls the interpolation between these two cases, while $ \gamma $ parametrises the type of extreme value distribution from which the underlying random variables are drawn from. For $ \alpha = 0 $ the Fr\'echet density applies, whereas for $ \alpha = 1 $ we identify a particular Fox H-function, which are a natural extension of hypergeometric functions into the realm of fractional calculus. In contrast for intermediate $ \alpha $ an entirely new function appears, which is not one of the extensions to the hypergeometric function considered to date. We derive series, integral and continued fraction representations of this latter function.