Proof
(related to Proposition: Antidifferences are Unique Up to a Periodic Constant)
- Assume, for a complex-valued function $f:\mathbb C\to\mathbb C$, the indefinite sum has two different values $G(x)=\sum f(x)$ and $F(x)=\sum f(x)$.
- By definition of the indefinite sum, $\Delta F(x)=f(x)$ and $\Delta G(x)=f(x)$.
- The both difference operators are therefore equal $\Delta F(x)=\Delta G(x)$.
- It follows from the basic calculations involving the difference operator $\Delta(F(x)-G(x))=0.$
- If we set $P(x)=F(x)-G(x)$, then $\Delta P(x)=0$, or, by the definition of the difference operator, $P(x+1)=P(x)$.
- Thus, $F(x)=\sum f(x)+P(x)$, $G(x)=\sum f(x)-P(x)$.
- In other words, any antidifference of $f$ can be written as its indefinite sum plus/minus a periodic constant $P(x).$
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References
Bibliography
- Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
