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

TL;DR

This paper establishes a coefficient-based criterion: if the entire series $\sum_{n=0}^ fty n!\,c_n\,z^n$ exists with non-terminating $c_n$, then the associated Maclaurin series $f(z)=\sum_{n=0}^\infty c_n\,z^n$ must have infinitely many zeros. The method combines Hadamard factorization for order-one entire functions with a precise description of derivative values $f^{(n)}(0)=k^n Q(n)$ to derive a contradiction from the assumed finiteness of zeros. This yields alternative proofs that Le Roy functions $f_r(z)=\sum_{n=0}^\infty z^n/(n!)^r$ for $\operatorname{Re}(r)>1$ and Bessel functions $J_\alpha(z)$ (for $\alpha\in\mathbb{R}$) possess infinitely many zeros. Overall, the work provides a simple, coefficient-driven criterion for zero-richness of power series with broad applications to classical special functions.

Abstract

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.

Theorems & Definitions (11)

• Theorem 1.1
• Lemma 1.2
• Lemma 1.3
• proof
• Remark 1.4
• proof
• proof
• Theorem 3.1
• proof
• Theorem 3.2