In this paper we obtain a recurrence relation between the pdf's (probability density functions) of order statistics. This relation is then extended to their cdf's (cumulative distribution functions) and to the expected values of a given function of each order statistic and a function of pairs of them. The interrelation between these results and other known formulas is shown. The application of the results to the preparation of tables of cdf's, expected values, variances, etc., of order statistics is indicated, and the numerical errors propagated in such computations analysed. Finally, an illustration is given. It is hoped that the unified approach presented here will provide greater insight and flexibility to procedures of numerical evaluation of expected values, etc., of order statistics.
"Recurrence Relations Between the PDF's of Order Statistics, and Some Applications." Ann. Math. Statist. 33 (1) 169 - 177, March, 1962. https://doi.org/10.1214/aoms/1177704721