SUMMARY Given a renewal situation specified by a 'lifetime' distribution function F(t), let H(t) be the renewal function and qS.(t) the nth 95-moment. Then two (integral) recurrence equations are developed for OSn. The first expresses 0. in terms of On-, and H, and the second is an integral equation involving On, On-, and F. It is then shown that if H(t) can be represented by an integral function of tm (for some m > 0), then so can Obn(t) for any n. Further, the coefficients in the series expansion of qS.(t) (in powers of tm) may be calculated either from the coefficients in the series expansion for F(t), or that for H(t).