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：我们探索密度函数 $ h(x;\ gamma,\ alpha) $, $x = in (0,\ infty) $, $\ gamma > 0, $ = gamma > 0, $ < alpha < 1 美元, 这来自一个吸引问题的域, 用于在超模和随机变量总和之间进行统计间插。 参数 $\ alpha$ 控制了这两个案例之间的内插, 而 $\ gamma $ 的对齐值显示的极端值分布类型, 并且从中抽取随机变量。 对于 $ \ alpha = 0 $ Fr\\' echet 密度适用, $ ALpha = 0, 而对于 $ = ALpha = 1 美元, 我们确定了特定的福克斯 H 函数, 这是超地球物理函数在微积积层的自然延伸。 。 相对于中间 $\alpha $ 完全新的函数, 似乎似乎不是迄今考虑的扩展函数的一部分。