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Proposition: Basic Calculations Involving Indefinite Sums
The indefinite sum is linear, i.e. for a complex-valued function $f:\mathbb C\to C$ and a constant $\lambda\in\mathbb C$ the following properties are fulfilled:
- $\sum \lambda f(x)=\lambda \sum f(x).$
- $\sum (f(x)\pm g(x))=\sum f(x)\pm \sum g(x).$
Moreover, for functions $f,g:\mathbb C\to C,$ the partial summation formula holds:
3. $\sum g(x)\Delta f(x)=f(x)g(x)-\sum f(x+1)\Delta g(x)$
- The linearity holds in analogy to the linearity of the integral.
- The partial summation formula is yet another formulation of the Abelian summation by parts, and will be proved here again using the methods of difference calculus.
- In the equations 1-3, the periodic constants are omitted, just like constants are omitted for the analogous formulas for the integral.
Table of Contents
Thank you to the contributors under CC BY-SA 4.0!
- Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
- Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
- Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960