Weierstrass products, canonical products, and genus

Let $(a_n{)}_{n\ge 1}$ be a sequence of nonzero complex numbers with no finite accumulation point, and let $(p_n{)}_{n\ge 1}$ be integers with $p_n\ge 0$.

Finite zero multisets are allowed as a separate degenerate case: the associated products have only finitely many factors, the empty product is $1$, and their canonical genus is defined to be $0$.

The product

${\prod }_{n\ge 1}{E}_{p_n}(z\mathrm{/}{a}_n)$

is a Weierstrass product for the zero sequence $(a_n)$.

If one integer $p\ge 0$ is used for every factor, so the product is

${\prod }_{n\ge 1}{E}_p(z\mathrm{/}{a}_n),$

it is the canonical product of genus $p$ associated to $(a_n)$. If there is at least one integer $p\ge 0$ for which that canonical product converges, the least such $p$ is the canonical genus of the sequence. If no such integer exists, its canonical genus is $+\mathrm{\infty }$.