Interpolation with entire function

Question

Is there any simple way to construct an entiere function $f$ such that : $$\forall p \in {\mathbb N} \quad f(2^p)=(-1)^p$$

Answer 1

Typically, one uses both the Weierstrass factorization and Mittag-Leffler theorem to prove the existence of an entire function $f$ such that $f(z_n)=w_n$ for given $z_n\to\infty$ and given $w_n$.

1. Get an entire function $g$ with a simple zero at every $z_n$ (by Weierstrass).
2. Get a meromorphic function $h$ with principal part $w_n(g'(z_n)(z-z_n))^{-1}$ at every $z_n$ (by Mittag-Leffler)
3. Let $f=gh$: this is an entire function and $f(z_n)=w_n$.

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But if you want to avoid heavy-duty theorems, see the article On Entire Function Interpolation by I. M. Sheffer, American Journal of Mathematics Vol. 49, No. 3 (Jul., 1927), pp. 329-342 which offers a more elementary proof of the existence of an entire function with prescribed values at positive integers.

Answer 2

By Weierstrass products, for each integer $k$ we can find an entire function $f_k$ such that $f_k(2^j)=0$ if $k\neq j$ and $f_k(2^k)=(-1)^k2^k$. Define $f:=\sum_{k=0}^{+\infty}2^{-k}f_k$. Using the relation about elementary factors, we can see that the convergence of this series is uniform over compact sets. Hence this defines an entire function which does the job.