User:Dom walden/Multivariate Analytic Combinatorics/Basics

Vectors in multidimensional space
The spaces $$\C^d$$ and $$\R^d$$ are made up of ordered sets of $$n$$ complex or real (resp.) numbers


 * $$\textbf{z} = (z_1, \cdots, z_d) \quad (z_i \in \C)$$


 * $$\textbf{x} = (x_1, \cdots, x_d) \quad (x_i \in \R)$$

We write $$\textbf{z} \in \C^d$$ or $$\textbf{x} \in \R^d$$.

Polydisk and polytorus
For the vectors $$\textbf{a} \in \C^d$$ and $$\textbf{r} \in \R_{>0}^d$$ the open polydisk centred at $$\textbf{a}$$ of radius $$\textbf{r}$$


 * $$D_\textbf{a}(\textbf{r}) = \{ \textbf{z} \in \C^d : |z_1 - a_1| < r_1, \cdots, |z_d - a_d| < r_d \}$$

and the polytorus


 * $$T_\textbf{a}(\textbf{r}) = \{ \textbf{z} \in \C^d : |z_1 - a_1| = r_1, \cdots, |z_d - a_d| = r_d \}$$

Multivariate Cauchy coefficient formula
This is the multivariate version of the Cauchy coefficient formula.

By the multiple Cauchy integral


 * $$f(z) = \frac{1}{(2\pi i)^n} \int_T \frac{f(\zeta) d\zeta_1 \cdots d\zeta_n}{(\zeta_1 - z) \cdots (\zeta_n - z)} = \frac{1}{(2\pi i)^n} \int_T \frac{f(\zeta) d\zeta}{(\zeta - z)}$$

and the fact that


 * $$\frac{1}{\zeta - z} = \sum_{|k|=0}^\infty \frac{(z - a)^k}{(\zeta - a)^{k+1}}$$

we can re-write $$f(z)$$


 * $$f(z) = \sum_{|k|=0}^\infty \frac{1}{(2\pi i)^n} \int_T \frac{f(\zeta) d\zeta}{(\zeta - a)^{k+1}} (z - a)^k = \sum_{|k|=0}^\infty c_k (z - a)^k$$

where:


 * $$c_k = \frac{1}{(2\pi i)^n} \int_T \frac{f(\zeta) d\zeta}{(\zeta - a)^{k+1}}$$

Domain of convergence
The domain of convergence of a power series is the set of points $$\textbf{z} \in \C^d$$ such that the power series converges absolutely for some neighbourhood of $$\textbf{z}$$.

The associated or conjugate radii of convergence are the vectors $$\textbf{r} \in \R_{>0}^d$$ such that the power series converges in the domain $$\{ \textbf{z} \in \C^d : |z_1 - a_1| < r_1, \cdots, |z_d - a_d| < r_d \}$$ and diverges in the domain $$\{ \textbf{z} \in \C^d : |z_1 - a_1| > r_1, \cdots, |z_d - a_d| > r_d \}$$. Note that $$\textbf{r}$$ is not necessarily unique and there may be infinite such $$\textbf{r}$$.

In our example, $$\frac{1}{1 - x - y}$$, the denominator is zero for $$0 \leq x \leq 1$$ and $$y = 1 - x$$.

Multivariate formal power series
For a generating function in $$d$$ variables.


 * $$\textbf{z} = (z_1, \cdots, z_d) \in \C^d$$, $$\textbf{n} = (n_1, \cdots, n_d) \in \N^d$$ and $$\textbf{z}^\textbf{n} = z_1^{n_1} \cdots z_d^{n_d}$$

The multivariate formal power series


 * $$F(\textbf{z}) = \sum_{\textbf{n} \in \N^d} f_\textbf{n} \textbf{z}^\textbf{n} = \sum_{(n_1, \cdots, n_d) \in \N^d} f_{n_1, \cdots, n_d} z_1^{n_1} \cdots z_d^{n_d}$$

For example


 * $$\frac{1}{1 - x - y} = \sum_{(n, m) \in \N^2} \binom{n + m}{m} x^m y^n = \sum_{n \geq 0} \sum_{m \geq 0} \binom{n + m}{m} x^m y^n$$

Diagonals
The central diagonal of a power series


 * $$\Delta F(\textbf{z}) = \sum_{n \geq 0} f_{n, \cdots, n} z^n$$

For our example power series, $$\frac{1}{1 - x - y}$$, we can represent it in two dimensions like below. The central diagonal is highlighted in green.

We can generalise this to a diagonal along a ray r, where $$\textbf{r} = (r_1, \cdots, r_d) \in \N^d$$


 * $$\Delta^{\textbf{r}} F(\textbf{z}) = \sum_{n \geq 0} f_{nr_1, \cdots, nr_d} z^n$$

For example


 * $$\Delta^{(2, 1)} \frac{1}{1 - x - y} = \sum_{n \geq 0} \binom{2n + n}{n} z^n$$

which is represented below in green.