# User:Dom walden/Multivariate Analytic Combinatorics/Basics

## Complex analysis in several variables

### Vectors in multidimensional space

The spaces ${\displaystyle \mathbb {C} ^{d}}$ and ${\displaystyle \mathbb {R} ^{d}}$ are made up of ordered sets of ${\displaystyle n}$ complex or real (resp.) numbers

${\displaystyle {\textbf {z}}=(z_{1},\cdots ,z_{d})\quad (z_{i}\in \mathbb {C} )}$
${\displaystyle {\textbf {x}}=(x_{1},\cdots ,x_{d})\quad (x_{i}\in \mathbb {R} )}$

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

### Polydisk and polytorus

For the vectors ${\displaystyle {\textbf {a}}\in \mathbb {C} ^{d}}$ and ${\displaystyle {\textbf {r}}\in \mathbb {R} _{>0}^{d}}$ the open polydisk centred at ${\displaystyle {\textbf {a}}}$ of radius ${\displaystyle {\textbf {r}}}$[1]

${\displaystyle D_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|

and the polytorus

${\displaystyle T_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|=r_{1},\cdots ,|z_{d}-a_{d}|=r_{d}\}}$

### Multivariate Cauchy coefficient formula

This is the multivariate version of the Cauchy coefficient formula.

By the multiple Cauchy integral[2]

${\displaystyle f(z)={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta _{1}\cdots d\zeta _{n}}{(\zeta _{1}-z)\cdots (\zeta _{n}-z)}}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -z)}}}$

and the fact that[3]

${\displaystyle {\frac {1}{\zeta -z}}=\sum _{|k|=0}^{\infty }{\frac {(z-a)^{k}}{(\zeta -a)^{k+1}}}}$

we can re-write ${\displaystyle f(z)}$

${\displaystyle f(z)=\sum _{|k|=0}^{\infty }{\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}(z-a)^{k}=\sum _{|k|=0}^{\infty }c_{k}(z-a)^{k}}$

where:

${\displaystyle c_{k}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}}$

### Domain of convergence

The domain of convergence of a power series is the set of points ${\displaystyle {\textbf {z}}\in \mathbb {C} ^{d}}$ such that the power series converges absolutely for some neighbourhood of ${\displaystyle {\textbf {z}}}$.[4]

The associated[5] or conjugate radii of convergence[6] are the vectors ${\displaystyle {\textbf {r}}\in \mathbb {R} _{>0}^{d}}$ such that the power series converges in the domain ${\displaystyle \{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}| and diverges in the domain ${\displaystyle \{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|>r_{1},\cdots ,|z_{d}-a_{d}|>r_{d}\}}$. Note that ${\displaystyle {\textbf {r}}}$ is not necessarily unique and there may be infinite such ${\displaystyle {\textbf {r}}}$.

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

## Multivariate generating functions

### Multivariate formal power series

For a generating function in ${\displaystyle d}$ variables.[7][8]

${\displaystyle {\textbf {z}}=(z_{1},\cdots ,z_{d})\in \mathbb {C} ^{d}}$, ${\displaystyle {\textbf {n}}=(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}}$ and ${\displaystyle {\textbf {z}}^{\textbf {n}}=z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}}$

The multivariate formal power series

${\displaystyle F({\textbf {z}})=\sum _{{\textbf {n}}\in \mathbb {N} ^{d}}f_{\textbf {n}}{\textbf {z}}^{\textbf {n}}=\sum _{(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}}f_{n_{1},\cdots ,n_{d}}z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}}$

For example[9]

${\displaystyle {\frac {1}{1-x-y}}=\sum _{(n,m)\in \mathbb {N} ^{2}}{\binom {n+m}{m}}x^{m}y^{n}=\sum _{n\geq 0}\sum _{m\geq 0}{\binom {n+m}{m}}x^{m}y^{n}}$

### Diagonals

The central diagonal of a power series[10]

${\displaystyle \Delta F({\textbf {z}})=\sum _{n\geq 0}f_{n,\cdots ,n}z^{n}}$

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

${\displaystyle x^{0}}$ ${\displaystyle x^{1}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{4}}$
${\displaystyle y^{0}}$ ${\displaystyle 1x^{0}y^{0}}$ ${\displaystyle 1x^{1}y^{0}}$ ${\displaystyle 1x^{2}y^{0}}$ ${\displaystyle 1x^{3}y^{0}}$ ${\displaystyle 1x^{4}y^{0}}$
${\displaystyle y^{1}}$ ${\displaystyle 1x^{0}y^{1}}$ ${\displaystyle 2x^{1}y^{1}}$ ${\displaystyle 3x^{2}y^{1}}$ ${\displaystyle 4x^{3}y^{1}}$ ${\displaystyle 5x^{4}y^{1}}$
${\displaystyle y^{2}}$ ${\displaystyle 1x^{0}y^{2}}$ ${\displaystyle 3x^{1}y^{2}}$ ${\displaystyle 6x^{2}y^{2}}$ ${\displaystyle 10x^{3}y^{2}}$ ${\displaystyle 15x^{4}y^{2}}$
${\displaystyle y^{3}}$ ${\displaystyle 1x^{0}y^{3}}$ ${\displaystyle 4x^{1}y^{3}}$ ${\displaystyle 10x^{2}y^{3}}$ ${\displaystyle 20x^{3}y^{3}}$ ${\displaystyle 35x^{4}y^{3}}$
${\displaystyle y^{4}}$ ${\displaystyle 1x^{0}y^{4}}$ ${\displaystyle 5x^{1}y^{4}}$ ${\displaystyle 15x^{2}y^{4}}$ ${\displaystyle 35x^{3}y^{4}}$ ${\displaystyle 70x^{4}y^{4}}$

We can generalise this to a diagonal along a ray r,[11] where ${\displaystyle {\textbf {r}}=(r_{1},\cdots ,r_{d})\in \mathbb {N} ^{d}}$

${\displaystyle \Delta ^{\textbf {r}}F({\textbf {z}})=\sum _{n\geq 0}f_{nr_{1},\cdots ,nr_{d}}z^{n}}$

For example

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

which is represented below in green.

${\displaystyle x^{0}}$ ${\displaystyle x^{1}}$ ${\displaystyle x^{2}}$ ${\displaystyle x^{3}}$ ${\displaystyle x^{4}}$
${\displaystyle y^{0}}$ ${\displaystyle 1x^{0}y^{0}}$ ${\displaystyle 1x^{1}y^{0}}$ ${\displaystyle 1x^{2}y^{0}}$ ${\displaystyle 1x^{3}y^{0}}$ ${\displaystyle 1x^{4}y^{0}}$
${\displaystyle y^{1}}$ ${\displaystyle 1x^{0}y^{1}}$ ${\displaystyle 2x^{1}y^{1}}$ ${\displaystyle 3x^{2}y^{1}}$ ${\displaystyle 4x^{3}y^{1}}$ ${\displaystyle 5x^{4}y^{1}}$
${\displaystyle y^{2}}$ ${\displaystyle 1x^{0}y^{2}}$ ${\displaystyle 3x^{1}y^{2}}$ ${\displaystyle 6x^{2}y^{2}}$ ${\displaystyle 10x^{3}y^{2}}$ ${\displaystyle 15x^{4}y^{2}}$
${\displaystyle y^{3}}$ ${\displaystyle 1x^{0}y^{3}}$ ${\displaystyle 4x^{1}y^{3}}$ ${\displaystyle 10x^{2}y^{3}}$ ${\displaystyle 20x^{3}y^{3}}$ ${\displaystyle 35x^{4}y^{3}}$
${\displaystyle y^{4}}$ ${\displaystyle 1x^{0}y^{4}}$ ${\displaystyle 5x^{1}y^{4}}$ ${\displaystyle 15x^{2}y^{4}}$ ${\displaystyle 35x^{3}y^{4}}$ ${\displaystyle 70x^{4}y^{4}}$

## Notes

1. Melczer 2021, pp. 94.
2. Shabat 1992, pp. 18.
3. Shabat 1992, pp. 19.
4. Melczer 2021, pp. 100.
5. Fuks 1963, pp. 46.
6. Shabat 1992, pp. 32.
7. Melczer 2021, pp. 93.
8. Mishna 2020, pp. 56.
9. Mishna 2020, pp. 142-145.
10. Mishna 2020, pp. 56-57.
11. Mishna 2020, pp. 57.

## References

• Fuks, B. A. (1963). Theory of Analytic Functions of Several Complex Variables. American Mathematical Society, Providence, Rhode Island.
• Melczer, Stephen (2021). An Invitation to Analytic Combinatorics: From One to Several Variables (PDF). Springer Texts & Monographs in Symbolic Computation.
• Mishna, Marni (2020). Analytic Combinatorics: A Multidimensional Approach. Taylor & Francis Group, LLC.
• Shabat, B. V. (1992). Introduction to Complex Analysis. Part II: Functions of Several Variables. American Mathematical Society, Providence, Rhode Island.