Proposition: Antidifferences are Unique Up to a Periodic Constant

If the indefinite sum $\sum f(x)$ exists, then it is unique, except a periodic constant, i.e. a complex-valued function $P:\mathbb C\to\mathbb C$ with the property $P(x+1)=P.$

In other words, every indefinite sum in the form $$F(x)=\sum f(x)+P(x)$$ is unique.

Notes

• This proposition is stated analogously to the proposition about antiderivatives are unique up to a constant.
• Thus, the periodic constant $P(x)$ plays the same role for the indefinite sum as the "constant" plays for the integral.

Proofs: 1

Propositions: 1

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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