How can we recursively implement a causal discrete-time truncated infinite impulse response (TIIR) filter with an arbitrary truncated polynomial impulse response:
$$h[n] = \begin{cases}\displaystyle\sum_{k=0}^K c_k n^k&\text{if }0\le n<N,\\ 0&\text{otherwise,}\end{cases}$$
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.