I'm currently reading a non-math paper that contains the following system of recursive equations: \begin{equation} x_{n+1}-x_{n+2} = c (x_n- x_{n+1}), n \in \mathbb{Z} \end{equation} where all $x_n$ and $c$ are real numbers. There is no initial value given. The paper then says that this is "Euler" and easily seen to be solved by $x_{n+1}=cx_n$. Indeed it is easily seen that this is a solution, but in what sense is this equation Euler?
2026-04-07 07:44:36.1775547876
A "Euler" difference equation
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The splution of $$X_{n+1}-X_{n+2}=c(X_n-X_{n+1})\implies Y_{n+1}=c Y_n \implies Y_n= c^n D$$ $$\implies X_{n}-X_{n+1} =c^n D$$ by telescopinc summation we get $$x_1-X_{n+1}=c\frac{c^n-1}{c-1} D\implies X_n=x_1-\frac{c^n-c}{c-1}$$
Sorry I don't really know if it is called Euler equation. But I will not be surprised if it is true. Why doesn't the OP share the context with us.