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

We explore the analytic properties of the density function $h(x;\gamma ,\alpha )$, $x\in (0,\mathrm{\infty })$, $\gamma >0$, $0<\alpha <1$ 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.