Insert the name of the directory where the downloaded files are located.
It takes a little while to include the following package (probably less than one minute):

SetDirectory[NotebookDirectory[]];
<<gammaReduce`

A special symbol greL is used to represent the summand, given by a product of ratios of Gamma functions.
By applying greLToSum, the ordinary expression is restored, where the summation indices are denoted by greX[i].

greL[{{},{},{},1},{{1},{3/2},{},1},{{1},{3/2},{},1},{{1/2},{3},{},1}]//greLToSum

greL[{{},{},{},1},{{(Subscript[a, 1]),...,(Subscript[a, L])},{(Subscript[b, 1]),...,(Subscript[b, L])},{},1},{{(Subscript[c, 1]),...,(Subscript[c, M])},{(Subscript[d, 1]),...,(Subscript[d, M])},{},1},{{(Subscript[e, 1]),...,(Subscript[e, N])},{(Subscript[f, 1]),...,(Subscript[f, N])},{},1}]=
[(∏_p)((Γ(n+(Subscript[a, p])))/(Γ(n+(Subscript[b, p]))))][(∏_q)((Γ(m+(Subscript[c, q])))/(Γ(m+(Subscript[d, q]))))][(∏_r)((Γ(n+m+(Subscript[e, r])))/(Γ(n+m+(Subscript[f, r]))))],
where the empty entries are kept for future usage.

To evaluate the sum (∑_(n,m=0)^∞), we can use greEvaluateSCL.

greEvaluateSCL[greL[{{},{},{},1},{{1},{3/2},{},1},{{1},{3/2},{},1},{{1/2},{3},{},1}]]

The following example is an 8 Gamma case.

greL[{{},{},{},1},{{1,1/2},{3/2,3/2},{},1},{{1},{3/2},{},1},{{1/2},{3},{},1}]//greLToSum

Timing[
greEvaluateSCL[greL[{{},{},{},1},{{1,1/2},{3/2,3/2},{},1},{{1},{3/2},{},1},{{1/2},{3},{},1}]]
]

We can also evaluate the sum of the expansion coefficients.
greSC works just as the Mathematica command “SeriesCoefficient” for greL.
For example:

greSC[greL[{{},{},{},1},{{1+r},{(3/2)},{},1},{{1},{(3/2)},{},1},{{(1/2)},{3},{},1}],{r,0,1}]

represents

 SeriesCoefficient[((Gamma[1+k+r] Gamma[1+m] Gamma[(1/2)+k+m])/(Gamma[(3/2)+k] Gamma[(3/2)+m] Gamma[3+k+m])),{r,0,1}]=((Gamma[1+k+r] Gamma[1+m] Gamma[(1/2)+k+m])/(Gamma[(3/2)+k] Gamma[(3/2)+m] Gamma[3+k+m]))PolyGamma[k+r]

greEvaluateSCL can also evaluate these sums.
In fact, greEvaluateSCL takes only  gre[...]   or  greSC[greL[...]]   as a valid argument.

greEvaluateSCL[greSC[greL[{{},{},{},1},{{1+r},{3/2},{},1},{{1},{3/2},{},1},{{1/2},{3},{},1}],{r,0,1}]]

Evaluation time increases rapidly, at higher order and with the number of Gamma functions.

greL[{{},{},{},1},{{1/2,1},{1,2},{},1},{{1,1},{3/2,2},{},1},{{1,3/2,1},{3/2,5/2,2},{},1}]//greLToSum

Timing[
greEvaluateSCL[
greL[{{},{},{},1},{{1/2,1},{1,2},{},1},{{1,1},{3/2,2},{},1},{{1,3/2,1},{3/2,5/2,2},{},1}]]
]

Timing[
greEvaluateSCL[greSC[greL[{{},{},{},1},{{1},{5/2+a},{},1},{{1},{5/2},{},1},{{3/2,2+e},{2,3+d},{},1}],{a,0,0},{e,0,0},{d,0,0}]]
]

Timing[
greEvaluateSCL[greSC[greL[{{},{},{},1},{{1},{5/2+a},{},1},{{1},{5/2},{},1},{{3/2,2+e},{2,3+d},{},1}],{a,0,1},{e,0,0},{d,0,0}]]
]

Timing[
greEvaluateSCL[greSC[greL[{{},{},{},1},{{1},{5/2+a},{},1},{{1},{5/2},{},1},{{3/2,2+e},{2,3+d},{},1}],{a,0,2},{e,0,0},{d,0,0}]]
]

Timing[
greEvaluateSCL[greSC[greL[{{},{},{},1},{{1},{5/2+a},{},1},{{1},{5/2},{},1},{{3/2,2+e},{2,3+d},{},1}],{a,0,2},{e,0,1},{d,0,0}]]
]