The empty product (x)_{0} is defined to be 1 in both cases.
The falling factorial occurs in a formula which represents polynomials using the forward difference operator Δ and which is formally similar to Taylor's theorem of calculus. In this formula and in many other places, the falling factorial (x)_{k} in the calculus of finite differences plays the role of x^{k} in differential calculus. Note for instance the similarity of
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