Dealing with Repeated Poles Analytically

A pole of multiplicity $m_i$ has $m_i$ coefficients^7.9 associated with it. For example,

$$H(z) \isdef \frac{7 - 5z^{-1}+ z^{-2}}{\left(1-\frac{1}{2}z^{-1}\right)^3}$$

$$= \frac{1}{\left(1-\frac{1}{2}z^{-1}\right)^3} + \frac{2}{\left(1-\frac{1}{2}z^{-1}\right)^2} + \frac{4}{\left(1-\frac{1}{2}z^{-1}\right)} \protect$$ (7.12)

and the three coefficients associated with the pole $z=1/2$ are 1, 2, and 4.

Let $r_{ij}$ denote the $j$ th coefficient associated with the pole $p_i$ , $j=1,\ldots,m_i$ . Successively differentiating $(1-p_iz^{-1})^{m_i}H(z)$ $k-1$ times with respect to $z^{-1}$ and setting $z=p_i$ isolates the coefficient $r_{ik}$ :

$$\begin{eqnarray*} r_{i1} &=& \left.(1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\ [5pt] r_{i2} &=& \left.\frac{1}{-p_i}\frac{d}{dz^{-1}} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\ [5pt] r_{i3} &=& \left.\frac{1}{2(-p_i)^2}\frac{d^2}{d(z^{-1})^2} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}\\ [5pt] r_{i4} &=& \left.\frac{1}{3\cdot 2(-p_i)^3}\frac{d^3}{d(z^{-1})^3} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i} \end{eqnarray*}$$

or

$$\zbox {r_{ik} = \left.\frac{1}{(k-1)!(-p_i)^{k-1}}\frac{d^{k-1}}{d(z^{-1})^{k-1}} (1-p_iz^{-1})^{m_i}H(z)\right\vert _{z=p_i}}$$