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Let $(a_n)$ and $(b_n)$ be two sequences of real numbers. Under which conditions on them exist a real analytic function $f$ such that $(\forall n\in \mathbb{N}) f(n) = a_n$ and $(\forall n\in \mathbb{N}) f'(n) = b_n$?

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The answer is yes, but the only way I know how to do it at the moment is through complex analysis with the Mittag-Leffler and/or Weierstrass factorization theorems. In fact you can come up with an entire function that will do the above; if your $a_n,b_n$ are real and you want a real solution, you can take the real part of the entire function.

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