The integral
$$Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)=\frac{1}{2}\int\_{-1}^1 I\_0\left[\tfrac{1}{2}(\lambda\_1-\lambda\_2) (1-x)\right] I\_0\left[\tfrac{1}{2} (\lambda\_1+\lambda\_2)(1+x)\right]\,e^{\lambda\_3 x}\,dx,$$
with $\lambda\_1,\lambda\_3,\lambda\_3\in\mathbb{R}$ and $I\_0$ a Bessel function, arises in the evaluation of an [integral over SO(3).](https://mathoverflow.net/q/302249/11260) From that derivation it follows that $Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)$ is invariant upon interchange of $\lambda\_1$ and $\lambda\_3$. I am unable to rewrite this integral in a manifestly permutation-invariant form, can someone help me out?



Invariance of an integral over SO(3) under permutation of parameters
Using the variable change $x=2t-1$, the integral $Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)$ can be put into the form
\begin{equation}
 Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)=\int\_{0}^1 I\_0\left[(\lambda\_1-\lambda\_2)
(1-t)\right]
e^{-\lambda\_3(1-t)}I\_0\left[(\lambda\_1+\lambda\_2)t\right]\,e^{\lambda\_3 t}\,dt
\end{equation} 
or, alternatively, $$ Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)=Z(1)$$ where 
\begin{equation}
 Z(T)=\int\_{0}^T I\_0\left[(\lambda\_1-\lambda\_2) (T-t)\right]
e^{-\lambda\_3(T-t)}I\_0\left[(\lambda\_1+\lambda\_2)t\right]\,e^{\lambda\_3 t}\,dt
\end{equation}
$Z(t)$ can be considered as a convolution integral. Its Laplace transform is 
\begin{equation}
 \mathcal{L}[Z](s)=\frac{1}{\sqrt{(s+\lambda\_3)^2-(\lambda\_1-\lambda\_2)^2}}\frac{1}{\sqrt{(s-\lambda\_3)^2-(\lambda\_1+\lambda\_2)^2}}
\end{equation} 
The square of the denominator of this transform is a monic polynomial which roots are $\left\lbrace -\lambda\_3+\lambda\_1-\lambda\_2, -\lambda\_3-\lambda\_1+\lambda\_2, \lambda\_3-\lambda\_1-\lambda\_2,\lambda\_3+\lambda\_1+\lambda\_2\right\rbrace$. This set is invariant upon interchange of the parameters. Thus, $Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)$ must share the same property. If an integral form needed, $Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)$ can be written as the inverse Laplace transform of this function which is permutation invariant:
$$Z\_3(\lambda\_1,\lambda\_2,\lambda\_3)=\frac 1{2i\pi}\int\_{c-i\infty}^{c+i\infty}\frac{e^s\,ds}{\sqrt{P(s)}}$$
where $c>\max(r\_1,r\_2,r\_3,r\_4)$, $r\_i$ are the roots given above and 
$$P(s)=s^4-2s^2(\lambda\_1^2+\lambda\_2^2+\lambda\_3^2)-8s\lambda\_1\lambda\_2\lambda\_3+(\lambda\_1-\lambda\_2-\lambda\_3)(\lambda\_2-\lambda\_3-\lambda\_1)(\lambda\_3-\lambda\_1-\lambda\_2)(\lambda\_1+\lambda\_2+\lambda\_3)
$$


