In analogy to the basic calculations involving derivatives, the following rules can be stated:

Proposition: Basic Calculations Involving the Difference Operator

Let $D\subseteq\mathbb R$ ($D$ being a subset of real numbers). Let $x, x+1\in D,\lambda\in\mathbb R,$ and let $f,g:D\to\mathbb R.$ Then

  1. $\Delta (\lambda f)(x)=\lambda \Delta f(x),$
  2. $\Delta(f\pm g)(x)=\Delta f(x)\pm \Delta g(x),$
  3. $\Delta (fg)(x)=g(x)\Delta f(x) + f(x+1)\Delta g(x)$ (also known as the product rule).
  4. If $g(x+1)g(x)\neq 0$ for all \(x\in D\), then (also known as the quotient rule): $$\Delta\left(\frac fg\right)(x)=\frac{g(x)\Delta f(x) - f(x)\Delta g(x)}{g(x+1)g(x)}.$$

Proofs: 1

Proofs: 1 2 3

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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