# 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:

1. $\sum \lambda f(x)=\lambda \sum f(x).$
2. $\sum (f(x)\pm g(x))=\sum f(x)\pm \sum g(x).$

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 1-3, the periodic constants are omitted, just like constants are omitted for the analogous formulas for the integral.

Proofs: 1

Proofs: 1

Github: ### References

#### Bibliography

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