# On Continuous Replicative Functions

I asked this question on math.stackexchange here, but it did not receive much attention. Thus, I was suggested to post it here.

Knuth, in The Art of Computer Programming Vol. 1, defines a replicative function as a function $f$ such that $$f(x)+f\left(x+\frac{1}{n}\right)+\cdots +f\left(x+\frac{n-1}{n}\right)=f(nx)$$ whenever $n\in\mathbf{Z}^{+}$. For $f(x)=\lfloor x\rfloor$, this is nothing but Hermite's identity. Knuth gives a few examples of replicative functions including the following.

• $f(x)=x-\frac{1}{2}$
• $f(x)=\begin{cases}1 && \text{if }x\in\mathbf{Z} \\ 0 && \text{otherwise}\end{cases}$
• $f(x)=\begin{cases}1 && \text{if }x\in\mathbf{Z}^{+} \\ 0 && \text{otherwise}\end{cases}$
• $f(x) = 1$ if there exists $r\in\mathbf{Q}$ and $m\in\mathbf{Z}$ such that $x=r\pi+m$, $f(x) = 0$ otherwise.
• The three other functions as above with $r$ and/or $m$ restricted to positive values.
• $f(x) = \log|2\sin(\pi x)|$, if the value $f(x)=-\infty$ is allowed.
• the sum of any two replicative functions.
• A constant multiple of a replicative functions.
• The function $g(x)=f(x-\lfloor x\rfloor)$, where $f$ is replicative.

Knuth then proposes to study the class of continuous replicative functions (the only one listed above is $f(x)=x-\frac{1}{2}$). However, as far as I can see, there seems to be a dearth of literature on this topic; the only piece I am aware of is here, but is not freely accessible.

Knuth also proposes to study the more general class of functions such that $$f(x)+f\left(x+\frac{1}{n}\right)+\cdots +f\left(x+\frac{n-1}{n}\right)=a_nf(nx)+b_n,$$ where $a_n$ and $b_n$ depend on $n$ but not $x$.

Is anyone aware of more information regarding this topic? Any help is greatly appreciated.

## 1 Answer

Functions satisfying such replicative identites are the cotangent, Bernoulli polynomials, and the logarithmic derivative of the Gamma function, to name a few.

The equation for n=2 is part of the Herglotz trick (see Aigner and Ziegler, the BOOK).

Replicative functions are discussed by Schroth (Aequ. Math. 20 (1980), 73-79), Jaeger (manusc. math. 56 (1986), 167-175), Walum (Pac. J. Math. 149 (1991), 383-396), Yoder (Aequat. Math. 13 (1975), 251-261).

The concept of distributions in this sense was developed by Kubert (Bull. Soc. Math. France 107 (1979), 179-202) and Lang; see also Milnor (Ens. Math. 29 (1983), 281-322).

Eisenstein's proof of the reciprocity law via the sine function essentially involves the replicative identity.

In physics, the map on the interval $[0,1]$ defined by $x \to 2x \bmod 1$ is called the Bernoulli shift, and its associated Perron-Frobenius operator is related to replicative functions.