Difference Operation. Condition: Alpha sub zero = Multiplicative Inverse of the Power

After doing a difference operation on a sequence with its coefficient alpha sub zero equal to the general term’s power, what the result will be?

With a short discussion of the Axiom of Choice...really what's stopping us from doing it that way?

Sources:

Gelfand, Gerver, Kirillov, Konstantinov and Kushnirenko - Sequences: 1969, problem 23