Constant Multiple of Replicative Function is Replicative

Theorem

Let $f: \R \to \R$ be a real function.

Let $f$ be a replicative function.

Let $c \in \R$ be a constant.

Let $g: \R \to \R$ be the real function defined as:

$\map g x = c \map f x$

Then $g$ is also a replicative function.

Proof

\(\ds \sum_{k \mathop = 0}^{n - 1} \map g {x + \frac k n}\)

\(=\)

\(\ds \sum_{k \mathop = 0}^{n - 1} c \map f {x + \frac k n}\)

\(\ds \)

\(=\)

\(\ds c \sum_{k \mathop = 0}^{n - 1} \map f {x + \frac k n}\)

\(\ds \)

\(=\)

\(\ds c \map f {n x}\)

Definition of Replicative Function

\(\ds \)

\(=\)

\(\ds \map g {n x}\)

Hence the result by definition of replicative function.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $39 \ \text{h)}$