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Let $f(x)$ be a function satisfying the functional equation $$ c_1(x)f(x)^2 + c_2(x) f(x) + c_3(x) f(x-1) + c_4(x) f(x+1) = 0, $$ where $c_1, \ldots, c_4$ are known functions. What can be said about $f$?

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    $\begingroup$ In general, nonlinear functional equations can't be solved in closed form. When they depend on arbitrary "known" functions, it's pretty much hopeless. Well, $f(x) = 0$ is a solution. $\endgroup$ – Robert Israel Jan 9 at 19:50
  • $\begingroup$ I could understand that someone views this question as too broad, or utterly hopeless. But voting "Not research level" seems pretty ridiculous. May be whatever information is known on the functions $c_i$ should be added to the question. $\endgroup$ – Jan-Christoph Schlage-Puchta Jan 13 at 11:20

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