This note expresses the first two moments of the order statistics in samples from the Weibull distribution (sometimes referred to as the "third" asymptotic distribution of extreme values) in terms of known (incomplete $B$ and $\Gamma$) functions. A similar procedure is applied to the "second" asymptotic distribution of extreme values.
"On Moments of Order Statistics from the Weibull Distribution." Ann. Math. Statist. 26 (2) 330 - 333, June, 1955. https://doi.org/10.1214/aoms/1177728551