The confluent multivariable Luaricella's hypergeometric function is defined as
$$\Phi^{(n)}_2\left(b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(b_1\right)_{m_1}\cdots\left(b_n\right)_{m_n}}{\left(c\right)_{m_1+\cdots+m_n}}\frac{x^{m_1}}{m_1!}\cdots \frac{x^{m_n}}{m_n!}$$
Where $(a)_m = \frac{\Gamma\left(a+m\right)}{\Gamma\left(a\right)} = a(a+1)\cdots (a+m-1)$ is the Pochhammer symbol for writing consecutive products. I am wondering is it possible to simplify the product
$$x^{n}\Phi^{(n)}_2\left(b_1,\cdots,b_n;c;\alpha_1 x,\cdots,\alpha_n x\right) =\\ \frac{1}{\prod_{i=1}^{n}\alpha_i} \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(b_1\right)_{m_1}\cdots\left(b_n\right)_{m_n}}{\left(c\right)_{m_1+\cdots+m_n}}\frac{(\alpha_1 x)^{m_1+1}}{m_1!}\cdots \frac{(\alpha_n x)^{m_n+1}}{m_n!}$$
and write it in terms of $\Phi^{(n)}_2\left(b_1,\cdots,b_n;c;\alpha_1 x,\cdots,\alpha_n x\right)$ (or its derivatives)?
Any hints are appreciated!
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We can further simplify the expression above to reach
$$\frac{1}{\prod_{i=1}^{n}\alpha_i} \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(b_1\right)_{m_1}\cdots\left(b_n\right)_{m_n}}{\left(c\right)_{m_1+\cdots+m_n}}\frac{(\alpha_1 x)^{m_1+1}}{m_1!}\cdots \frac{(\alpha_n x)^{m_n+1}}{m_n!}\\= \frac{(c-n)_{n}}{\prod_{i=1}^{n}(b_i-1)\alpha_i} \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(b_1-1\right)_{m_1+1}\cdots\left(b_n-1\right)_{m_n+1}}{\left(c-n\right)_{(m_1+1)+\cdots+(m_n+1)}}\frac{(\alpha_1 x)^{m_1+1}}{m_1!}\cdots \frac{(\alpha_n x)^{m_n+1}}{m_n!}\\$$
The only problem left is factorials in the denominator!
