In this paper, we establish several recurrence relations satisfiedby the single and product moment generating function of progressive first failure censoring from an right truncated exponential distribution. These relations may then be used, for example, several recurrence relations satisfiedby the single and product moment generating function of progressive first failure censoring from an right truncated exponential distribution and to compute all the means, variances and covariances of right truncated exponential distribution progressive first failure censoring for all sample sizes n and all censoring schemes (R_1,R_2,…,R_m), m < n. The results presented in the paper generalize the results given by Aggarwala and Balakrishnan [1] for the single moments and product moments of order statistics from the exponential and right truncated exponential distributions. To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moment generating function are established for progressive first failure censoring from the right truncated exponential distribution. Further, the results are specialized to the progressively type-II right censored order statistics (PTIIRCOS).