Recursive Term Computation

Our Lagrange interpolation filter is again

$$\hat{H}_\Delta(q^{-1}) \;=\;\sum_{n=0}^{N} \hbar _\Delta(n)(q-1)^n$$

where the coefficients of $(q-1)^n=(z^{-1}-1)^n$ are again

$$\hbar _\Delta(n) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\frac{1}{n!}\prod_{k=0}^{n-1}(\Delta-k) \;=\;\frac{\Delta(\Delta-1)(\Delta-2)\cdots(\Delta-n+1)}{n!}$$

Note that we can recursively compute the terms in the sum from left to right:

$$\begin{eqnarray*} \hbar _\Delta(n) &=& \hbar _\Delta(n-1)\cdot \frac{\Delta-n+1}{n}\\ (q-1)^{n} &=& (q-1)^{n-1}\cdot (q-1) \end{eqnarray*}$$

Thus, we can crank out the terms in series and sum the intermediate signals: