Entire Dixon functions

Dixon elliptic functions $\text{sm}\ z$ and $\text{cm}\ z$ are meromorphic functions, both having the same poles. This implies that
$$\text{sm}\ z=\frac{f(z)}{h(z)},$$ and
$$\text{cm}\ z=\frac{g(z)}{h(z)},$$ where $f$, $g$ and $h$ are entire functions, such that 

$$f^3+g^3=h^3.$$

$$f(x)=\sum_{n=0}^\infty \frac{\omega^n x^{3n}}{(3n)!}$$

$$g(x)=\sum_{n=0}^\infty \frac{\omega^n x^{3n+1}}{(3n+1)!}$$

$$h(x)=\sum_{n=0}^\infty \frac{\omega^n x^{3n+2}}{(3n+2)!}$$

where

$$\omega^3=1,$$

is the cube root of unity.