Recursive filter with truncated polynomial impulse response
I don't think you can, I don't see any special property of the truncated polynomial representation that would help here.

The impulse response coefficients are related to the polynomial coefficients through a set of linear equations, so we can simply write this as a matrix multiplication.

$$h = M \cdot c$$
where $h$ is the impulse response vector, $c$ the polynomial coefficient vector and $M$ a matrix that looks something like
$$M = \begin{bmatrix}
1 &amp; 0 &amp; 0 &amp; ...  \\ 
1 &amp; 1 &amp;1  &amp; ...  \\ 
1 &amp; 2 &amp; 4 &amp;  ... \\ 
1 &amp; ... &amp; ... &amp; ...
\end{bmatrix}$$

That matrix appears to be invertible, so we simply express the polynomial coefficients as 

$$c = M^{-1} \cdot h$$

for any given impulse response. That means that every FIR impulse response can be represented in truncated polynomial form and there is really nothing special about it. If there were a way to implement this recursively, it would mean that every FIR filter can be implemented recursively which, to the best of my knowledge, is not the case.