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Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
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Solutions of complex linear difference equations
I'm wondering what the solutions of complex linear difference equations like
\begin{equation}
f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C}
\end{equation}
look …