Hypergeometric function with a matrix argument

I am looking for the evaluation of a Hypergeometric function with a matrix argument as for example in Koev and Edelman or as showcased in this Wikipedia article.

From what I understand from Mathematica's documentation, it only accepts a scalar as the last argument.

Answer

Matrix functions in MMA

First of all MMA does in general not support matrix arguments in its standard functions. Therefore there are special functions MatrixExp and MatrixPower available. But, as will be shown in this answer, it is possible to create user defined functions via infinite series, and it turns out that MMA is surprisingly well able in dealing with these matrix functions. The idea is simple: a function with a known series expansion can be transformed into a Matrix function letting z^k -> MatrixPower[z,k].

Hypergeometric matrix function in one variable

Let us start with the well known function $2F1(a,b,c;z)$ which in MMA is Hypergeometric2F1[a,b,c,z], and define

matrix2F1[a_, b_, c_, z_] := 
 Sum[Pochhammer[a, k] Pochhammer[b, k]/
    Pochhammer[c, k] MatrixPower[z, k]/k!, {k, 0, \[Infinity]}]

Here z is a matrix.

Example 1: Pauli matrix

z = PauliMatrix[1]

$\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right)$

matrix2F1[a,b,c,z]

$\left( \begin{array}{cc} \frac{1}{2} (\text{Hypergeometric2F1}[a,b,c,-t]+\text{Hypergeometric2F1}[a,b,c,t]) & \frac{1}{2} (-\text{Hypergeometric2F1}[a,b,c,-t]+\text{Hypergeometric2F1}[a,b,c,t]) \\ \frac{1}{2} (-\text{Hypergeometric2F1}[a,b,c,-t]+\text{Hypergeometric2F1}[a,b,c,t]) & \frac{1}{2} (\text{Hypergeometric2F1}[a,b,c,-t]+\text{Hypergeometric2F1}[a,b,c,t]) \\ \end{array} \right)$

I found it very surprising that MMA computes the series and recognizes closed expressions without problems.

Example 2: general 2x2 matrix

z = {{p, q}, {r, s}}

$\left( \begin{array}{cc} p & q \\ r & s \\ \end{array} \right)$

matrix2F1[1, 2, 2, z t] // FullSimplify;

% // MatrixForm

$\left( \begin{array}{cc} \frac{1-s t}{1-t (s+q r t)+p t (-1+s t)} & -\frac{q t}{-1+t (p+s+q r t-p s t)} \\ -\frac{r t}{-1+t (p+s+q r t-p s t)} & \frac{1-p t}{1-t (s+q r t)+p t (-1+s t)} \\ \end{array} \right)$

Hypergeometric matrix function in two variables

For two variables there is a problem to be clarified first: two matrices in general do not commute, i.e. the result of a multiplication depends on the order. Hence $f(x,y)\neq f(y,x)$, in general. A relation between $f(x,y)$ and $f(y,x)$ become a Little simpler if x.y = A y.x + B with some scalars A and B. But let's us Abandon this question for a Moment and go to the hypergeometrics.

There are 4 different generalizations of the hypergeometric function, called Appell functions (http://mathworld.wolfram.com/AppellHypergeometricFunction.html).

Let's take the first one and define

matrixAppellF1[a_, b_, b1_, c_, x_, y_] := 
 Sum[(Pochhammer[a, m + n] Pochhammer[b, m] Pochhammer[b1, n])/(
   m! n! Pochhammer[c, m + n])

    MatrixPower[x, m].MatrixPower[y, n], {m, 0, \[Infinity]}, {n, 
   0, \[Infinity]}]

Here we have taken the powers of x together to the left of the powers of y. This seems to be Kind of "natural", but there are other possibilities.

Example 3: AppelF1[a,b,b1,c,x,y]

First check result with scalars

AppellF1[1, 1, 1, 1, t , u ]

$\frac{1}{(1-t) (1-u)}$

x = PauliMatrix[1];

y = PauliMatrix[3];

matrixAppellF1[1, 1, 1, 1, t x, u y]

$\left( \begin{array}{cc} -\frac{1}{(-1+t) (1+t) (1-u)} & -\frac{t}{(-1+t) (1+t) (1+u)} \\ -\frac{t}{(-1+t) (1+t) (1-u)} & -\frac{1}{(-1+t) (1+t) (1+u)} \\ \end{array} \right)$

Summary

We have shown that MMA is very well suited to handle complicated functions with Matrix arguments.We simply have to replace in the power series Expansion of the usual function the power of the variable z^k with MatrixPower[z,k].

Surprisingly enough, MMA can do the infinite sums and provide closed expressions.

By the way: the there is no need for the user to think about eigenvalues or diagonalization.

Best regards, Wolfgang