Probability generating function: Read Generating Functions.
Moment generating function: Read Moment-generating function - Wikipedia, the free encyclopedia.
The moment generating function of Y = aX + b is : See Moment Generating Function.
In your question and b = 0.
The moment generating function of a random variable distributed uniformly on (a, b) is : See Uniform distribution (continuous) - Wikipedia, the free encyclopedia.
Substitute a = 0 and b = 1 to get the answer your shooting for when finding ......