I have found the following equation may related to the modified Bessel function of the first kind, but how to prove it? The equation is as follows: \begin{align} &\frac{\Gamma(T)\exp\{-||y||^{2}\}}{\pi^{T}}\int_{0}^{\infty}\frac{\exp\{-z(\rho T+1)\}}{(||y||\sqrt{\rho Tz})^{T-1}}I_{T-1}(2||y||\sqrt{\rho Tz})dz\\ =& \frac{e^{-||y||^{2}/(1+\rho T)}||y||^{2(1-T)}\Gamma(T)}{\pi^{T}(1+\rho T)}\tilde{\gamma}(T-1,\frac{\rho T||y||^{2}}{1+\rho T})(1+\frac{1}{\rho T})^{T-1} \end{align} where $||y||$ denotes the $l_{2}$-norm of $y$, $\tilde{\gamma}(\cdot,\cdot)$ represents the lower incomplete gamma function and $I_{T-1}(\cdot)$ the modified Bessel function of the first kind.
How should I verify the above equation ?
Any useful suggestion is welcome.
Thanks a lot, Liu.