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If $\sum_n n! c_n z^n$ is entire and $c_n$ does not terminate, then $\sum_n c_n z^n$ has infinitely many zeros
Alann Rosas
https://arxiv.org/abs/2510.10054 https://

If $\sum_n n! c_n z^n$ is entire and $c_n$ does not terminate, then $\sum_n c_n z^n$ has infinitely many zeros
We prove that if $\sum_n n! c_n z^n$ is entire and $c_n$ does not terminate, then $\sum_n c_n z^n$ has infinitely many zeros. We then use this result to give alternative proofs that the Le Roy functions $f_r(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^r}$ for $r>1$ and Bessel functions $J_α(z)=\sum_{m=0}^\infty \frac{(-1)^m}{m!Γ(m+α+1)}\left(\frac{z}{2}\right)^{2m+α}$ for $α\in\mathbb R$ have infinitely many zeros.