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Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ such that $f\big(f(n)\big)=-(q-p)\,f(n)+pq\,n$ for all $n\in\mathbb{Z}$.

Let $p$ and $q$ be integers. Let $S$ be a subset of $\mathbb{Z}$, and $f:S\to S$. Consider the functional equation of the form $$f\big(f(n)\big)=-(q-p)\,f(n)+pq\,n\text{ for each }n\in S\,.\tag{*}$$ ...