I’m just learning. For all complex values except the negative integers, the harmonic number is given by
$$H_x = x\sum_{k=1}^\infty \frac{1}{k(x+k)}$$
$H_0=0$ and there are infinitely many roots at negative values.
Is this an entire function? If so, could we factor the harmonic number as a Weierstrass product? I could not find any sources on the harmonic number as a product.
Added: It seems that $\frac{H_x}{x!}$ should admit a product form, right?