Somehow Mathematica cannot evaluate the following with "{n,1,30}" instead of the "{n, 1.000000000001, 30.000000000001}", but it is safe to assume the sum over m,j,k would exist and be the trivariate Lauricella function if the pattern from the previous answer was continued.
Sum[((2.4/(4-3*I))^j*(2.4/(5+0.2*I))^k*(2.4/(-3))^m*(-3*(4-3*I)*(5+0.2*I))^(n/8)*(n/8)!*E^((1/8)*(2*{0,1,2,3,4,5,6,7} + 1)*Pi*I*n)*
Pochhammer[-(n/8), j]*Pochhammer[-(n/8), k]*Pochhammer[-(n/8), m]*
Pochhammer[1 - n, j + k + m])/(((2.4)^((7*n)/8 - 1)*(1 - (7*n)/8)!*n!)*
(j!*k!*m!*Pochhammer[2 - (7*n)/8, j + k + m])), {n, 1.000000000001, 30.000000000001},{m, 0, 30}, {j, 0, 30},
{k, 0, 30}]
Sum[((2.4/(4-3*I))^j*(2.4/(5+0.2*I))^k*(2.4/(-3))^m*(-3*(4-3*I)*(5+0.2*I))^(n/8)*(n/8)!*E^((1/8)*(2*{0,1,2,3,4,5,6,7} + 1)*Pi*I*n)*
Pochhammer[-(n/8), j]*Pochhammer[-(n/8), k]*Pochhammer[-(n/8), m]*
Pochhammer[1 - n, j + k + m])/(((2.4)^((7*n)/8 - 1)*(1 - (7*n)/8)!*n!)*
(j!*k!*m!*Pochhammer[2 - (7*n)/8, j + k + m])), {n, 1.000000000001, 30.000000000001},{m, 0, 30}, {j, 0, 30},
{k, 0, 30}]
Out[]=
{1.53951+1.36862,-0.0742334+1.98736,-1.29628+1.19492,-1.61212+0.130105,-1.50762-0.94037,-0.410105-2.03555,1.40796-1.60858,1.95289-0.0965094}
In[]:=
ListPlot[(Tooltip[{Re[#1],Im[#1]}]&)/@%17,AspectRatio1]
Out[]=
In[]:=
Solve[z^8+(z+2.4)(z-3)(z+4-3*I)(z+5+0.2*I)==0,z]
Out[]=
{{z-1.61212+0.130105},{z-1.50762-0.94037},{z-1.29628+1.19492},{z-0.410105-2.03555},{z-0.0742334+1.98736},{z1.40796-1.60858},{z1.53951+1.36862},{z1.95289-0.0965094}}
In[]:=
{-1.6121210956062624+0.1301045200460071,-1.5076205611886622-0.9403701903712678,-1.2962766986694525+1.1949203519413107,-0.41010471703586765-2.035550685071011,-0.07423335303260141+1.9873628524232323,1.407962609700967`-1.608577488647575,1.5395060852480682+1.3686200656624847,1.9528877305838106-0.09650942598318149`}
Out[]=
{-1.61212+0.130105,-1.50762-0.94037,-1.29628+1.19492,-0.410105-2.03555,-0.0742334+1.98736,1.40796-1.60858,1.53951+1.36862,1.95289-0.0965094}
In[]:=
ListPlot[(Tooltip[{Re[#1],Im[#1]}]&)/@%21,AspectRatio1]
Out[]=
Note:TheMathematicaStackExchangeLauricellafunctionroutinefromhttps://mathematica.stackexchange.com/questions/154434/evaluating-lauricella-functions-in-mathematicadoesnotworkhere?:
In[]:=
lauricellaTerm[m_Integer?Positive,a_,b_?VectorQ,c_,z_?VectorQ]:=(Pochhammer[a,m]/Pochhammer[c,m])*Sum[BellY[m,k,Table[(j-1)!b.z^j,{j,m}]],{k,m}]/m!
In[]:=
With[{a=2,b={1,2/3},c=3,z={1/5,2/5}},1+NSum[lauricellaTerm[m,a,b,c,z],{m,1,30}]]
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Out[]=
1+NSumlauricellaTermm,2,1,,3,,,{m,1,30}
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