◀ ▲ ▶Branches / Combinatorics / Proposition: Basic Calculations Involving Indefinite Sums
Proposition: Basic Calculations Involving Indefinite Sums
The indefinite sum is linear, i.e. for a complexvalued 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. $\sum g(x)\Delta f(x)=f(x)g(x)\sum f(x+1)\Delta g(x)$
Notes
 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 13, the periodic constants are omitted, just like constants are omitted for the analogous formulas for the integral.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", AddisonWesley, 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