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Joint moments involving arbitrary powers of order statistics are the main concern. Consider order statistics u1 ≤ u2 ≤ ··· ≤ uk coming from a simple random sample of size n from a real continuous population where u1 = xr(1):n is order-statistic #r1, u2 = xr(1)+r(2):n is order statistic #(r1 + r2), et al., and uk = xr(1)+···+r(k):n is order statistic #(r1 +···+ rk). Product moments are examined of the type E[u1α(1) · u2α(2)· ····ukα(k)] where α1, ..., αk are arbitrary quantities that might be complex numbers, and E[·] denotes the s-expected value. Some explicit evaluations are considered for a logistic population. Detailed evaluations of all integer moments of u1 and recurrence relations, recurring only on the order of the moments, are given. Connections to survival functions in survival analysis, hazard functions in reliability situations, real type-1, type-2 β and Dirichlet distributions are also examined. Arbitrary product moments for the survival functions are evaluated. Very general results are obtained which can be used in many problems in various areas.