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I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as: $$ f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i b_i e^{i(q_{i,x}x+q_{i,y}y)}} $$ where $z=x+iy$ and the (infinite set of) vectors $\{\vec{q_i}\}$ live on a quasi-lattice (that is, they can be spanned by a finite set of vectors with integer coefficients). Naturally, when $\{\vec{q_i}\}$ live on a lattice this defines an elliptic function.

Is there any literature concerning such functions (or any other similar definition of a quasicrystalline meromorphic function)? I would specifically be interested if there is anything that can be said about its poles\zeros.

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  • $\begingroup$ If $q_{i,x}, q_{i,y}$ and are real numbers, this function is not meromorphic (does not satisfy Cauchy-Riemann equations). If they are complex numbers then Cauchy-Riemann mean that $q_{i,x}=q_{i,y}$ for all $i$, so you have a singly periodic function. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 13, 2021 at 16:29
  • $\begingroup$ You never obtain an elliptic function by this kind of formula. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 13, 2021 at 16:30
  • $\begingroup$ @alexandre eremenko Why not? If f is elliptic I can write it as the quotient of two doubly periodic bounded functions, then the sums are simply the Fourier expansion $\endgroup$
    – Yarden Sheffer
    Commented Sep 13, 2021 at 18:22
  • $\begingroup$ Not analytic? An analytic function cannot be bounded in the plane. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 13, 2021 at 20:26

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