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.


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.


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