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.