Recursive filter with truncated polynomial impulse response How can we recursively implement a causal discrete-time truncated infinite impulse response (TIIR) filter with an arbitrary truncated polynomial impulse response:

$$h[n] = \begin{cases}\displaystyle\sum_{k=0}^K c_k n^k&amp;\text{if }0\le n&lt;N,\\
0&amp;\text{otherwise,}\end{cases}$$

where real $c_k$ are the arbitrary coefficients of the polynomial, nonnegative integer $K$ is the degree of the polynomial, $0^0 = 1$, and positive integer $N$ is the length of the part of the impulse response that may be non-zero?

Perhaps a recursive form allows time complexity to be not dependent on $N$, giving an efficient implementation when $K$ is small, even when $N$ is large. Moving average can be implemented recursively as an integrator with tail cancellation. That's a $K=0$ degree truncated polynomial impulse response. Perhaps higher degree polynomials are also possible.

A cascade of identical moving average filters gives a B-spline impulse response, which is a piece-wise polynomial of any desired degree but lacking arbitrary control of the polynomial coefficients. Perhaps some form of tail cancellation allows to isolate a single piece of a higher degree B-spline impulse response and to linearly combine the piece with the same of lower degree B-splines to get arbitrary polynomials of any degree. Or maybe there is another approach.

Some promising literature I found:
Oscar G. Ibarra-Manzano, Yuriy S. Shmaliy, "Implementation of Digital Unbiased FIR Filters with Polynomial Impulse Responses", Circuits Systems and Signal Processing, April 2012.
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.