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$\newcommand{\R}{\mathbb{R}}$
Suppose the next equation:
$$
\sum_{n \ge 0}^\infty a(n) ( \cosh{(x\cdot n))} - \sinh{(x\cdot n))}) = 0
$$
Is it possible that exists another solution distinct to $x = 0$ , $x \in \R $?
How can be this solution if exists?