Multidimensional Z-transform

Remarks

For $n = 1$ this is the bilateral Z-transform. The series is a Laurent series in several variables. Absolute convergence at $(z_1, \dots, z_n)$ depends only on $(\abs{z_1}, \dots, \abs{z_n})$.

Example: $x[k_1, k_2] = a^{k_1} b^{k_2}$ for $k_1, k_2 \ge 0$ and $x[k] = 0$ otherwise, with $a, b \in \C$. The sum is a product of two geometric series, and $X(z_1, z_2) = \frac{1}{(1 - a z_1^{-1})(1 - b z_2^{-1})}$ for $\abs{z_1} > \abs{a}$ and $\abs{z_2} > \abs{b}$.

Non-example: $x[k] = 1$ for all $k \in \Z^2$ has an empty region of convergence. For every $z$, choose signs $\epsilon_1, \epsilon_2 \in \set{1, -1}$ with $\abs{z_j}^{-\epsilon_j} \ge 1$ for $j = 1, 2$. Then the infinitely many terms with $k_j = \epsilon_j m_j$, $m_1, m_2 \ge 0$, all have absolute value at least $1$, so the series diverges.