$\newcommand{\lm}{\lambda}$ $\newcommand{\bR}{\mathbb{R}}$ Let $m$ be an integer $>1$. Define


$$ I\_m:\bR^m\to \bR,\;\; I\_m(\lm\_1,\dotsc, \lm\_m)=\int\_{S^{m-1}}\exp\Bigl(-\sum\_{j=1}^m \lm\_j^2x\_j^2\;\Bigr)\; |dA(x)|, $$


where $S^{m-1}$ is the unit sphere in $\bR^m$ and $|dA(x)|$ denotes the "area" element on $S^{m-1}$.


The function $I\_m$ is real analytic and symmetric in the variables $\lm\_1^2,\dotsc, \lm\_m^2$ and in fact it has a Taylor expansion


$$I\_m (\lm\_1,\dotsc, \lm\_m)=2\sum\_{h=0}^\infty\frac{(-1)^h}{\Gamma(\frac{m}{2}+h)}\sum\_{h\_1+\cdots+h\_m=h}\frac{\Gamma(h\_1+\frac{1}{2})\cdots \Gamma(h\_m+\frac{1}{2})}{h\_1!\cdots h\_m!} \lm\_1^{2h\_1}\cdots \lm\_m^{2h\_m}. $$


In particular, $I\_m$ can be expressed as a function of the symmetric polynomials


$$s\_k =\sum\_{j=1}^m \lm\_j^{2k},\;\; k=1,\dotsc, m. $$


**Question 1.** Is there a more compact description of $I\_m$ of the form


$$I\_m(\lm\_1,\dotsc, \lm\_m)=F\_m(s\_1,\dotsc, s\_m), $$


where $F\_m$ is some "*classical*" function?


**Question 2.** $\DeclareMathOperator{\diag}{Diag}$ $\DeclareMathOperator{\tr}{tr}$ Consider the symmetric matrix


$$\Lambda=\diag(\lm\_1,\dotsc, \lm\_m). $$


Is there some function $V\_m:\bR\to \bR$ such that


$$ I\_m(\lm\_1,\dotsc, \lm\_m)=e^{-\tr V\_m(\Lambda)} ? $$


Can one describe such a $V\_m$ explicitly? I'm vague about the term explicit, but I would be very pleased if $V\_m$ were a "special" function.


The second question may suggest the origin of $I\_m$. I stumbled onto $I\_m$ when I bumped into a certain ensemble of random real, symmetric $m\times m$ matrices. (The story is too bushy to include it here.) 



Can one recognize this symmetric function?
Your function is a special case of the [Confluent Hypergeometric function of matrix argument](http://dlmf.nist.gov/35.6), in particular it is ${}\_1F\_1(\frac{1}{2},\frac{n}{2},L)$, where $L$ is a diagonal matrix having $-\lambda\_i^2$ ($1\le i \le n$) as its diagonal components. These functions can be more generally represented using [Zonal Polynomials](http://sites.stat.psu.edu/~richards/talks/zphf.pdf).


This function arises as the normalization constant of a [Bingham Distribution](http://en.wikipedia.org/wiki/Bingham_distribution). More details about this function and other related hypergeometric functions can be found in the textbooks:


1. *Directional Statistics*, Mardia and Jupp (2000)
2. *Aspects of multivariate statistical theory*, Muirhead (1982).


