NOTE: I edited this question, following the comments of Alexander Eremenko and Paul Garrett.
I have a question concerning elliptic functions that maybe you can help me shed light on. I am a theoretical physicist so I excuse myself preemptively for any mathematical imprecision.
It is a fact that any two elliptic functions $f(z)$ and $g(z)$, having the same periodicities, satisfy an algebraic relation $$F[f,g]=0\;,$$ where $F$ is a polynomial in two variables with constant coefficients.
Similarly, it is a fact that, given an elliptic function $f(z)$, its convolution with another generic function $h(z)$ is again an elliptic function, having the same periodicities: $$c(z) = \intop_{-\infty}^{\infty} dw\,f(z-w)h(w)\;,$$ $$\left\lbrace\begin{array}{c} f(z+\omega_1) = f(z) \\ f(z+\omega_3) = f(z)\end{array}\right.\quad \Longrightarrow \quad \left\lbrace\begin{array}{c} c(z+\omega_1) = c(z) \\ c(z+\omega_3) = c(z)\end{array}\right.\;.$$ The function $h(z)$ is such that the convolution is well-defined.
Given the two facts above, I deduce that there exist an algebraic relation of the form $$\mathcal{F}[f,c] = \sum_{n=0}^N\sum_{m=0}^M \alpha_{n,m} f(z)^n\,c(z)^m = 0\;.$$
EDITS
As it was remarked in the comments, the above statement is, in general, not true. In fact, let $\omega_1$ and $\omega_3$ be the two periods of the elliptic function $f(z)$. Suppose moreover that $\omega_1\in\mathbb R_{>0}$ and $-i\omega_3\in\mathbb R_{>0}$ (Note that I use $\omega_{1,3}$ for the entire periods, not the half-ones as is customary). Let moreover be $\lbrace \zeta_k\rbrace$ be the poles of $f(z)$ in the fundamental rectangle with vertices $0$, $\omega_1$, $\omega_3$ and $\omega_1+\omega_3$. Finally, suppose that $-i\omega_3>\Im\zeta_k>0,\;\forall k$, so that the convolution is well-defined (also, $h(z)$ has no poles on the real axis). Then it seems to me quite evident that the convolution $c(z)$ is periodic with period $\omega_1$, since any performed change of integration variable would not move the integration contour. In formulae \begin{eqnarray*}c(z) &= \intop_{-\infty}^{\infty} dw\, f(w)h(z-w) = \intop_{-\infty}^{\infty} dw\, f(w-\omega_1)h((z+\omega_1)-w) =\\ &= \intop_{-\infty}^{\infty} dw\, f(w)h((z+\omega_1)-w) = c(z+\omega_1)\;. \end{eqnarray*}
On the other hand, the periodicity $z\rightarrow z+\omega_3$ is in general violated. In fact, the convolution will be multi-valued, as Alexander pointed out. We can say that $$ c(z) = \intop_{-\infty}^{\infty} dw\, f(z-w)h(w) = \intop_{-\infty}^{\infty} dw\, f(z-w+\omega_3)h(w) = c(z+\omega_3)\;,$$ owing to the periodicity of $f$. However we might as well compute as follows \begin{eqnarray*} c(z) &=& \intop_{-\infty}^{\infty} dw\, f(w)h(z-w) = \intop_{-\omega_3-\infty}^{-\omega_3+\infty} dw\, f(w)h(z-w) - 2\pi i\sum \textrm{Res}_{\zeta_k} = \\ &=& \intop_{-\infty}^{+\infty} dw\, f(w-\omega_3)h((z+\omega_3)-w) - 2\pi i\sum \textrm{Res}_{\zeta_k} = c(z+\omega_3) - 2\pi i\sum \textrm{Res}_{\zeta_k}\;. \end{eqnarray*} Here I denote schematically by $\sum \textrm{Res}_{\zeta_k}$ the sum over all the residues picked up by the shift in the integration contour.
As an example, I computed numerically $c(z)$ for $f(z) = \textrm{cn}_l(z)$ and $h(z) = e^{-z^2}$, verifying the $\omega_1$ periodicity. For the $\omega_3$ lack of periodicity, I see that the two definitions of the convolution yield different numerical results, but I cannot match the difference with the residues (I probably am counting them wrong).
So, given the above, I guess that my question below concerns a specific case in which $h(z)$ is periodic of period $\omega_3$. For example, the case $f(z) = \textrm{cn}_l(z)$ and $h(z) = \frac{1}{\cosh(\frac{\pi}{2K_{1-l}}z)}$, where $\omega_3 = 2i K_{1-l}$. For this situation, the convolution should be doubly periodic and my question should make sense.
I tried the example I mentioned and I verified that both definitions of the convolution (with $w-z$ in $f$ or $g$) yield the same result when shifted by the periods. Again I could not verify the nullity of the sum of residues, probably because I am doing something wrong.
To answer the latest comments: what I mean by convolution is a 1-dimensional integral over the real numbers.
QUESTION
My question is the following: is there a simple (or at the very least, feasible) way to find the integers $N$ and $M$ and, possibly, compute the coefficients $\alpha_{n,m}$?
Thanks a lot!
