Part: Discrete Calculus and Difference Equations

This part is dedicated to the so-called discrete calculus, sometimes also called the calculus of finite differences. This is a subject providing methods for solving difference equations. As opposed to differential equations, difference equations are useful as a mathematical representation for certain physical problems.

The purpose of this part is to develop a calculus of "differences" and a "sums", in analogy to the calculus of derivatives and Riemann integrals.

Examples: 1

  1. Definition: Difference Operator
  2. Proposition: Basic Calculations Involving the Difference Operator
  3. Proposition: Nth Difference Operator
  4. Proposition: Difference Operator of Powers
  5. Definition: Factorial Polynomials
  6. Definition: Falling and Rising Factorial Powers of Functions
  7. Theorem: Taylor's Formula Using the Difference Operator
  8. Definition: Indefinite Sum, Antidifference
  9. Proposition: Basic Calculations Involving Indefinite Sums
  10. Proposition: Antidifferences of Some Functions
  11. Definition: Falling And Rising Factorial Powers
  12. Proposition: Difference Operator of Falling Factorial Powers
  13. Theorem: Fundamental Theorem of the Difference Calculus

Chapters: 1
Solutions: 2

Thank you to the contributors under CC BY-SA 4.0!




  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