# hypergeomatrix
## Evaluation of the hypergeometric function of a matrix argument (Koev & Edelman's algorithm)
Let $(a\_1, \ldots, a\_p)$ and $(b\_1, \ldots, b\_q)$ be two vectors of real or
complex numbers, possibly empty, $\alpha > 0$ and $X$ a real symmetric or a
complex Hermitian matrix.
The corresponding *hypergeometric function of a matrix argument* is defined by
$${}\_pF\_q^{(\alpha)} \left(\begin{matrix} a\_1, \ldots, a\_p \\\\ b\_1, \ldots, b\_q\end{matrix}; X\right) = \sum\_{k=0}^{\infty}\sum\_{\kappa \vdash k} \frac{{(a\_1)}\_{\kappa}^{(\alpha)} \cdots {(a\_p)}\_{\kappa}^{(\alpha)}} {{(b\_1)}\_{\kappa}^{(\alpha)} \cdots {(b\_q)}\_{\kappa}^{(\alpha)}} \frac{C\_{\kappa}^{(\alpha)}(X)}{k!}.$$
The inner sum is over the integer partitions $\kappa$ of $k$ (which we also
denote by $|\kappa| = k$). The symbol ${(\cdot)}\_{\kappa}^{(\alpha)}$ is the
*generalized Pochhammer symbol*, defined by
$${(c)}^{(\alpha)}\_{\kappa} = \prod\_{i=1}^{\ell}\prod\_{j=1}^{\kappa\_i} \left(c - \frac{i-1}{\alpha} + j-1\right)$$
when $\kappa = (\kappa\_1, \ldots, \kappa\_\ell)$.
Finally, $C\_{\kappa}^{(\alpha)}$ is a *Jack function*.
Given an integer partition $\kappa$ and $\alpha > 0$, and a
real symmetric or complex Hermitian matrix $X$ of order $n$,
the Jack function
$$C\_{\kappa}^{(\alpha)}(X) = C\_{\kappa}^{(\alpha)}(x\_1, \ldots, x\_n)$$
is a symmetric homogeneous polynomial of degree $|\kappa|$ in the
eigen values $x\_1$, $\ldots$, $x\_n$ of $X$.
The series defining the hypergeometric function does not always converge.
See the references for a discussion about the convergence.
The inner sum in the definition of the hypergeometric function is over
all partitions $\kappa \vdash k$ but actually
$C\_{\kappa}^{(\alpha)}(X) = 0$ when $\ell(\kappa)$, the number of non-zero
entries of $\kappa$, is strictly greater than $n$.
For $\alpha=1$, $C\_{\kappa}^{(\alpha)}$ is a *Schur polynomial* and it is
a *zonal polynomial* for $\alpha = 2$.
In random matrix theory, the hypergeometric function appears for $\alpha=2$
and $\alpha$ is omitted from the notation, implicitely assumed to be $2$.
Koev and Edelman (2006) provided an efficient algorithm for the evaluation
of the truncated series
$$\sideset{\_p^m}{\_q^{(\alpha)}}F \left(\begin{matrix} a\_1, \ldots, a\_p \\\\ b\_1, \ldots, b\_q\end{matrix}; X\right) = \sum\_{k=0}^{m}\sum\_{\kappa \vdash k} \frac{{(a\_1)}\_{\kappa}^{(\alpha)} \cdots {(a\_p)}\_{\kappa}^{(\alpha)}} {{(b\_1)}\_{\kappa}^{(\alpha)} \cdots {(b\_q)}\_{\kappa}^{(\alpha)}}
\frac{C\_{\kappa}^{(\alpha)}(X)}{k!}.$$
Hereafter, $m$ is called the *truncation weight of the summation*
(because $|\kappa|$ is called the weight of $\kappa$), the vector
$(a\_1, \ldots, a\_p)$ is called the vector of *upper parameters* while
the vector $(b\_1, \ldots, b\_q)$ is called the vector of *lower parameters*.
The user has to supply the vector $(x\_1, \ldots, x\_n)$ of the eigenvalues
of $X$.
For example, to compute
$$\sideset{\_2^{15}}{\_3^{(2)}}F \left(\begin{matrix} 3, 4 \\\\ 5, 6, 7\end{matrix}; 0.1, 0.4\right)$$
you have to enter
```haskell
hypergeomat 15 2 [3.0, 4.0], [5.0, 6.0, 7.0] [0.1, 0.4]
```
We said that the hypergeometric function is defined for a real symmetric
matrix or a complex Hermitian matrix $X$. Thus the eigenvalues of $X$
are real. However we do not impose this restriction in `hypergeomatrix`.
The user can enter any list of real or complex numbers for the eigenvalues.
### Gaussian rational numbers
The library allows to use **Gaussian rational numbers**, i.e. complex numbers
with a rational real part and a rational imaginary part. The Gaussian rational
number $a + ib$ is obtained with `a +: b`, e.g. `(2%3) +: (5%2)`. The imaginary
unit usually denoted by $i$ is represented by `e(4)`:
```haskell
ghci> import Math.HypergeoMatrix
ghci> import Data.Ratio
ghci> alpha = 2%1
ghci> a = (2%7) +: (1%2)
ghci> b = (1%2) +: (0%1)
ghci> c = (2%1) +: (3%1)
ghci> x1 = (1%3) +: (1%4)
ghci> x2 = (1%5) +: (1%6)
ghci> hypergeomat 3 alpha [a, b] [c] [x1, x2]
26266543409/25159680000 + 155806638989/3698472960000*e(4)
```
### Univariate case
For $n = 1$, the hypergeometric function of a matrix argument is known as the
[generalized hypergeometric function](https://mathworld.wolfram.com/HypergeometricFunction.html).
It does not depend on $\alpha$. The case of $\sideset{\_{2\thinspace}^{}}{\_1^{}}F$ is the most known,
this is the Gauss hypergeometric function. Let's check a value. It is known that
$$\sideset{\_{2\thinspace}^{}}{\_1^{}}F \left(\begin{matrix} 1/4, 1/2 \\\\ 3/4\end{matrix}; 80/81\right) = 1.8.$$
Since $80/81$ is close to $1$, the convergence is slow. We compute the truncated series below
for $m = 300$.
```haskell
ghci> h <- hypergeomat 300 2 [1/4, 1/2] [3/4] [80/81]
ghci> h
1.7990026528192298
```
## References
- Plamen Koev and Alan Edelman.
*The efficient evaluation of the hypergeometric function of a matrix argument*.
Mathematics of computation, vol. 75, n. 254, 833-846, 2006.
- Robb Muirhead.
*Aspects of multivariate statistical theory*.
Wiley series in probability and mathematical statistics.
Probability and mathematical statistics.
John Wiley & Sons, New York, 1982.
- A. K. Gupta and D. K. Nagar.
*Matrix variate distributions*.
Chapman and Hall, 1999.