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Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$,

$$f(\omega z) = \omega^k f(z)$$

for all $z\in U \subseteq \mathbb C$. (Obviously the domain $U$ has to be closed under multiplying with $\omega$.)

For $f$ analytic around $0$, this is equivalent to: In the Taylor expansion $f(z) = \sum a_m z^m$, all $a_m$ except possibly those with $m \equiv k \pmod n$ are zero.

The sum of two functions of type $k \pmod n$ is again of type $k \pmod n$. The product of a function of type $k \pmod n$ and a function of type $l \pmod n$ is of type $(k+l) \pmod n$. Also, the derivative of a differentiable function of type $k \pmod n$ is of type $k-1 \pmod n$.

For $n=2$, this retrieves the classical "even" and "odd functions" taught to this day in high schools.

I just stumbled upon these, or shifts of them, as rare exceptional solutions to functional identities which otherwise usually have no interesting solutions. I wondered if there is a better name for them, and if they are useful in some theory I have missed.

(This seems to be at least one level of difficulty below modular forms of weight $k$, but maybe somebody can take me by the hand and show a connection to those. Or to the seemingly very different generalization here, maybe bringing in representation theory of the cyclic group $\mathbb Z/n$.)

Torsten Schoeneberg
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  • What is interesting about odd or even functions that you want generalized? – Conrad May 14 '24 at 14:27
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    @Conrad: Fair question, but not my perspective. I am rather hoping for a richer theory in the general setting which might not show yet in the special case $n=2$. [As in: Somebody asks, "I noticed that beyond complex conjugation on $\mathbb C | \mathbb R$, one could look at any field extension $L|K$ and the $L$-automorphisms which fix $K$. Is there something interesting to say about that?" Then "What is interesting about complex conjugation that you want generalized?" is not the perspective that leads to Galois theory, Kummer theory etc.] – Torsten Schoeneberg May 14 '24 at 18:07
  • That being said, if there is a property of such $f$ as mapping that generalizes the classical "symmetric around y-axis / through the origin", that would be neat. My feeling is we're secretly looking at Riemann surfaces which are $n$-fold coverings of $U$ here, and have $\mathbb Z/n$ operating on that, but I'm just far too ignorant in complex analysis to make that precise and/or fruitful. – Torsten Schoeneberg May 14 '24 at 18:10
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    I think the most natural characterization is that they are the functions of the form $z^k g(z^n)$, as alluded to in the post. This directly generalizes what we know about analytic odd/even functions, and there's surely a way of phrasing it in terms of the $n$-fold cover implied by the inner function $z^n$. – Greg Martin May 14 '24 at 21:14
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    In general the symmetry above is usually used to construct functions of lower order but not sure otherwise if is really of much interest (eg $\cos$ even means $\cos \sqrt z$ is entire order $1/2$ and if we symmetrize more - say to roots of order $4$ and do the appropriate composition we get a function order $1/4$ etc – Conrad May 14 '24 at 23:11

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