I want to add up the following two Fox H-functions $$ H_{1,2}^{\,1,1} \!\left[ -\lambda^2 \left|x\right|^{2\alpha^\prime} \left| \begin{matrix} ( 0 , 1 ) \\ ( 0 , 1 ) & ( 0 , 2\alpha^\prime ) \end{matrix} \right. \right] + i\times H_{1,2}^{\,1,1} \!\left[ -\lambda^2 \left|x\right|^{2\alpha^\prime} \left| \begin{matrix} ( \frac{1}{2} , 1 ) \\ ( \frac{1}{2} , 1 ) & ( 0 , 2\alpha^\prime ) \end{matrix} \right. \right] $$
where $\lambda$ and $\alpha^\prime$ are real numbers and
$$ H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} \right. \right] = \frac{1}{2\pi i}\int_L \frac {(\prod_{j=1}^m\Gamma(b_j+B_js))(\prod_{j=1}^n\Gamma(1-a_j-A_js))} {(\prod_{j=m+1}^q\Gamma(1-b_j-B_js))(\prod_{j=n+1}^p\Gamma(a_j+A_js))} z^{-s} \, ds $$
Is it possible to convert the above summation to just one Fox H-function?
We have $$H = H_{1, 2}^{1, 1} {\left(-z \middle| {(0, 1) \atop (0, 1), (0, \alpha)} \right)} + i H_{1, 2}^{1, 1} {\left(-z \middle| {(\frac 1 2, 1) \atop (\frac 1 2, 1), (0, \alpha)} \right)} = \\ \frac 1 {2 \pi i} \int_{\mathcal L} \frac {\Gamma(1 + y) \Gamma(-y) + i \Gamma(\frac 1 2 + y) \Gamma(\frac 1 2 - y)} {\Gamma(1 + \alpha y)} (-z)^y dy = \\ -\frac 1 {2 \pi i} \frac 1 \pi \int_{\mathcal L} \frac {\Gamma(y) \Gamma(1 - y) \Gamma(\frac 1 2 + y) \Gamma(\frac 1 2 - y)} {\Gamma(1 + \alpha y)} e^{-i \pi y} (-z)^y dy.$$ For $z > 0$, $e^{-i \pi y} (-z)^y = z^y$. Therefore $$H = -\frac 1 \pi H_{2, 3}^{2, 2} {\left(z \middle| {(1, 1), (\frac 1 2, 1) \atop (1, 1), (\frac 1 2, 1), (0, \alpha)} \right)}.$$