Completely monotonic integer degrees for a class of special functions

Let f(n)(x) (n = 0, 1, . . .) be the remainders for the asymptotic formula of ln Gamma(x) and R-n(x) = (-1)(n) f(n)(x). This paper introduced the concept of completely monotonic integer degree and discussed the ones for the functions (-1)(m) R-n((m))(x), then demonstrated the correctness of the existing conjectures by using a elementary simple method. Finally, we propose some operational conjectures which involve the completely monotonic integer degrees for the functions (-1)(m) R-n((m))(x) for m = 0, 1, 2, . . ..