integral of definite Meijer's G-function

Question

I am working on my research and i have one integral seems difficult to me given as: $$\int_0^\infty x^{-\omega}\exp(-\theta x)\large{G}_{1,2}^{1,1} \left( \beta x^{\alpha/2} \left| \begin{array}{cc} 1 \\ \zeta,0 \end{array} \right. \right) \, dx$$ where $\alpha,\omega,\theta$ are positive numbers (variable), i know how to solve it when the Meijer argument is $x$ only not $x^{\alpha/2}$. could someone help me on that. Thanks in advance!

Answer 1

Integrating a product of two G-functions gives a Fox H-function if only one of the G-functions is linear: $$\int_0^\infty x^{-\omega} G_{0, 1}^{1, 0}\left( \theta x \middle| { - \atop 0} \right) G_{1, 2}^{1, 1}\left( \beta x^{\alpha/2} \middle| { 1 \atop \zeta, 0} \right) dx = \theta^{\omega - 1} H_{2, 2}^{1, 2} \left( \beta \theta^{-\alpha/2} \middle| {(1, 1), (\omega, \alpha/ 2) \atop (\zeta, 1), (0, 1)} \right), \\ \alpha, \beta, \theta > 0, \quad \omega < 1 + \frac {\alpha \zeta} 2.$$ The result can be converted to a G-function when $\alpha$ is rational.