On the Density arising from the Domain of Attraction between Sum and Supremum: the α-Sun operator
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We explore the analytic properties of the density function h(x;γ,α), x ∈ (0,∞), γ > 0, 0 < α < 1 which arises from the domain of attraction problem for a statistic interpolating between the supremum and sum of random variables. The parameter α controls the interpolation between these two cases, while γ parametrises the type of extreme value distribution from which the underlying random variables are drawn from. For α = 0 the Fréchet density applies, whereas for α = 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 α 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.
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