# Need a Solution for a Definite Integral that involves Meijer G-function

I am trying to get the solution for the integral given below

$$I = \int_0^\infty t^{-1} \exp(At)\exp(-Bt^2) \large{G}_{0,2}^{2,0}\left( Ct \left| \begin{array}{cc} - \\ \alpha, \beta \end{array} \right. \right) \ dt$$

where A, B, and C are constants, and $\alpha$ and $\beta$ are positive parameters. It's known that the $\exp$ function can be written in the form of Miejer G-function by the using the relation given as

$$\exp(t)= \large{G}_{0,1}^{1,0}\left( -t \left| \begin{array}{cc} - \\ 0 \end{array} \right. \right)$$

Any help will be appreciated. Thanks

Not an answer, but an expression for the integral which may be easier to handle: from the Wolfram function site or from (9.34.3) in Gradshteyn and Rydzhik, $${G}_{0,2}^{2,0}\left( t \left| \begin{array}{cc} - \\ \alpha, \beta \end{array} \right. \right) =2t^{\tfrac{\alpha+\beta}{2}}K_{\alpha-\beta}\left( 2\sqrt{t} \right)$$ where $K_\nu(.)$ is the modified Bessel function. The integral becomes (with $a=A/C$, $b=B/C^2$) $$I=2\int_0^\infty \exp(-bt^2+at)K_{\alpha-\beta}\left( 2\sqrt{t} \right) t^{\tfrac{\alpha+\beta}{2}-1}\,dt$$