The following theorem for the discrete difference operator corresponds to a similar theorem from the continuous calculus, called the Taylor's formula.

Theorem: Taylor's Formula Using the Difference Operator

Let $f:D\to\mathbb C$ be a factorial polynomial with a suitable domain $D\subseteq\mathbb C$. Using the falling factorial powers and the nth difference operator, the value at $f(x)$ can be written as

$$f(x)=f(0)+\sum_{k=1}^n \frac{\Delta^k f(0)}{k !} x^{\underline{k}}.$$

Proofs: 1

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  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
  2. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
  3. Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960