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Out of curiosity, we're looking for a function $φ$ such that $(φ ∘ φ)(x) = ax^2 + bx + c$.

For $x^2$, a trivial function is $φ(x) = x^{√2}$. Indeed: $(x^{√2})^{√2} = x^{√2 × √2} = x^2$. Is there a general expression for any polynomial of degree 2?

Do you know where to find documentation about recursive functionals? I haven't found anything on Wikipedia.

pronodingo
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 13 '23 at 14:44
  • Imo, to clarify your post and make it focus on a single question, you would just have to suppress the last paragraph (about "recursive functionals"). – Anne Bauval Jul 13 '23 at 14:50

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