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We explicitly obtain the distributions $p_k^\\alpha(t)$, the probability generating functions $G_\\alpha(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\\ uniform random variables. 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We explicitly obtain the distributions $p_k^\\alpha(t)$, the probability generating functions $G_\\alpha(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\\ uniform random variables. 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