Discrete distributions from moment generating function

The recovering of positive discrete distributions from their moment generating function (mgf) is considered. From mgf and some integer moments, proper fractional moments are obtained. The latter represent the available information of the distribution. Then maximum entropy machinery is invoked to find the approximate distribution. It is proved that the approximant converges in entropy, in information divergence and then in total variation, so that accurate expected values may be obtained. Some numerical experiments are illustrated.