A hypergeometric single sum, like a Mittag Leffler function uses the Pochhammer symbol $(a)_n$ multiplication formula to easily have a univariate hypergeometric function $_p\text F_q$ closed form:
$$\sum_{n=0}^\infty\frac{z^n}{(1)_{kn}}=\sum_{n=0}^\infty\frac{(k^{-k}z)^n}{n!\prod\limits_{p=1}^{k-1}\left(\frac pk\right)_n}=\,_0\text F_{k-1}\left(\frac1k,\dots,\frac{k-1}k;\frac z{k^k}\right)\tag1$$
Similarly, if a hypergeometric double sum, with indices $m,n$, and summand including $\Gamma(jm+kn+a);j,k\in\Bbb N$ can be written with $\Gamma(m+n+b),\Gamma(m+b),\Gamma(n+b)$ instead, then it would have a bivariate hypergeometric function closed form. However, the Gauss multiplication formula:
$$\Gamma(jm+kn+a)=k^{k+mj+a-\frac12}(2\pi)^\frac{1-k}2\prod_{p=0}^{k-1}\Gamma\left(n+\frac{jm+a+p}k\right)\tag 2$$
only gives $\Gamma(jm+kn+a)$ in terms of $\Gamma(n+rm+c),r\in\Bbb Q$. Using Pochhammer symbol addition formulas:
$$\Gamma(jm+kn+a)=\Gamma(a)(a)_{jm+kn}=\Gamma(a)(a)_{jm}(a+jm)_{kn}$$
and its multiplication formula has same problem as $(2)$ preventing a closed form for the double sum.
Is there any to write $\Gamma(jm+kn+a)$ in terms of $\Gamma(m+n+b),\Gamma(m+b),\Gamma(n+b)$ or an analogous method, like in $(1)$, but for a double sum?
Not an answer just some representation for the Gamma function (which could help) :
We have $x$ a positive integer :
$$\Gamma(x+1)=-\sum_{n=2}^{\infty}\int_{0}^{\infty}\frac{x!}{n!\left(x-n\right)!}\frac{e^{-t}}{t}\left(e^{-t}-1\right)^{n-1}dt$$
See https://arxiv.org/pdf/2301.09699.pdf for other representations
Reference :
https://en.wikipedia.org/wiki/Dixon%27s_identity