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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. $\Delta(f\pm g)(x)=\Delta f(x)\pm \Delta g(x),$
  2. $\Delta (\lambda f)(x)=\lambda \Delta f(x),$
  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 the quotient rule):
    $$\Delta\left(\frac fg\right)(x)=\frac{g(x)\Delta f(x) – f(x)\Delta g(x)}{g(x+1)g(x)}.$$

| | | | | created: 2020-03-23 21:20:40 | modified: 2020-03-24 09:20:06 | by: bookofproofs | references: [1112], [8404], [8405]

1.Proof: (related to "Basic Calculations Involving the Difference Operator")

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[8405] Bool, George: “A Treatise on the Calculus of Finite Differences”, Dover Publications, Inc., 0

[1112] Graham L. Ronald, Knuth E. Donald, Patashnik Oren: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition

[8404] Miller, Kenneth S.: “An Introduction to the Calculus of Finite Differences And Difference Equations”, Dover Publications, Inc, 0