TITLE: Invariance of an integral over SO(3) under permutation of parameters
QUESTION [5 upvotes]: 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). 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?

REPLY [7 votes]: 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 
$$P(s)=s^4-2s^2(\lambda_1^2+\lambda_2^2+\lambda_3^2)-8s\lambda_1\lambda_2\lambda_3+(\lambda_1-\lambda_2-\lambda_3)(\lambda_2-\lambda_3-\lambda_1)(\lambda_3-\lambda_1-\lambda_2)(\lambda_1+\lambda_2+\lambda_3)
$$