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

Question

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

Answer 1

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, \begin{equation} {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) \end{equation} where $K_\nu(.)$ is the modified Bessel function. The integral becomes (with $a=A/C$, $b=B/C^2$) \begin{equation} I=2\int_0^\infty \exp(-bt^2+at)K_{\alpha-\beta}\left( 2\sqrt{t} \right) t^{\tfrac{\alpha+\beta}{2}-1}\,dt \end{equation}

Answer 2

This is not an answer, but a hint. Your Integral may be expressed as a Meijer G-Function of two Variables, which is discussed by Agarwal in 1964 for the first time. A more general form is the H-Fox-Function of two Variables which can be found in the standard publication of Mathai: https://www.researchgate.net/publication/266566090_The_H-function_Theory_and_Applications. An example of how to calculate your integral can be found here: https://www.researchgate.net/publication/269504875_Capacity_of_k_-_m_Shadowed_Fading_Channels. You may find also the source code for implementing the function in Mathematica in the publication above, I already implemented it, too, but it works only for one variable.