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.

翻译：我们探讨了密度函数$ h(x;\gamma,\alpha) $的解析性质，其中$x \in (0,\infty)$，$\gamma > 0$，$0 < \alpha < 1$，该函数来源于随机变量的上确界与和之间插值统计量的吸引域问题。参数$\alpha$控制着这两种情形之间的插值，而$\gamma$则参数化了基础随机变量所服从的极值分布类型。当$\alpha = 0$时，适用Fr\'echet密度；当$\alpha = 1$时，我们识别出一个特定的Fox H函数，该函数是超几何函数向分数阶微积分领域的自然推广。相比之下，对于中间的$\alpha$值，出现了一个全新的函数，它不属于迄今为止考虑过的超几何函数的任何推广形式。我们推导了后一种函数的级数、积分和连分式表示。