Recursive form representation of FIR filter
The frequency sampling representation of a length $N$ FIR filter's transfer function

$$H(z)=\frac{1-z^{-N}}{N}\sum_{k=0}^{N-1}\frac{\tilde{H}[k]}{1-e^{j2\pi k/N}z^{-1}}\tag{1}$$

is a systematic way of finding a recursive implementation for any FIR filter. The coefficients $\tilde{H}[k]$ in $(1)$ are the DFT coefficients of the impulse response $h[n]$. I do not know of any other such systematic way.

However, there are special cases for which a recursive implementation can be found by using standard finite sum identities. One such class of finite impulse response transfer functions is

$$H(z)=K\sum_{n=0}^{N-1}a^nz^{-n}=K\frac{1-a^Nz^{-N}}{1-az^{-1}}\tag{2}$$

A special case of $(2)$ is the moving average filter

$$H(z)=\frac{1}{N}\sum_{n=0}^{N-1}z^{-n}=\frac{1}{N}\frac{1-z^{-N}}{1-z^{-1}}\tag{3}$$