# Product of Weierstrass elementary factors is entire

From specialfunctionswiki

## Theorem

Let $\{a_n\}$ denote a sequence of complex numbers with $a_n \neq 0$ and $\displaystyle\lim_{n\rightarrow\infty} |a_n| = \infty$. If $\{p_n\}$ is a sequence of nonnegative integers such that for every $r>0$ $$\displaystyle\sum_{k=1}^{\infty} {r \choose |a_k|}^{1+p_n} < \infty,$$ where ${r \choose r_k}$ denotes a Binomial coefficient, then the product $$P(z) = \displaystyle\prod_{k=1}^{\infty} E_{p_k} \left( \dfrac{z}{a_k} \right),$$ where $E_{p_k}$ denotes a Weierstrass elementary factor defines an entire function which has a zero at each $a_k$ and no other zeros. If a complex number $w$ appears $n$ times in the sequence $\{a_n\}$, then $P$ has a zero of order $m$ at $w$.