Study on expansion and properties of grey accumulating generation operator

By utilizing Gamma function expanded for integer factorial, this paper deduces the analytical expression of integer order accumulating generation operator based on the one order accumulating generation operator, then expands the integer order accumulating generation operator into the fractional order accumulating generation operator, and gives the analytical expression of fractional order accumulating generation operator. Actually, one order accumulating generation operator and integer order accumulating generation operator are both special cases of the fractional order accumulating generation operator. Theoretical proof and numerical simulation proves that the fractional order accumulating generation operator satisfies commutative law and exponential law. Study on the fractional order accumulating generation operator would help develop grey prediction model with the fractional order operators and would widen the application fields of the grey prediction model.