In[]:=
$Version
Out[]=
14.2.0 for Microsoft Windows (64-bit) (December 26, 2024)
In[]:=
F[a_,Var_]:=NIntegratey*Exp-
2
2
a
z(2
2
z
-1)
Var
,{y,0,1},{z,0,y}
In[]:=
F
1
10
,
1
100

Out[]=
0.449737
In[]:=
G[a_,Var_,M_]:=NSum
1
3
(-11+m)
2
-2+
4m
3
a
-2m/3
Var
2/3
2
Var-Gamma1+
m
3
+Gamma1+
m
3
,
4
2
a
Var
+4
4/3
a
1/3
Var
Gamma
1+m
3
-Gamma
1+m
3
,
4
2
a
Var

3m!
,{m,0,M}
In[]:=
G
1
10
,
1
100
,20
Out[]=
0.449737
In[]:=
H[a_,Var_,M_]:=NSum
m
-
4
2
a
Var
n
2
2
a
Var
(1+3m+n)(3+3m+n)m!n!
,{m,0,M},{n,0,M}
In[]:=
H
1
10
,
1
100
,20
Out[]=
0.449737
H1[a_,Var_,M_]:=NSum
Pochhammer[1,3m+n]Pochhammer[3,3m+n]
3Pochhammer[2,3m+n]Pochhammer[4,3m+n]
*
m
-
4
2
a
Var
n
2
2
a
Var
m!n!
,{m,0,M},{n,0,M}(*ThisseriescanbyexpressedbySrivastava-Daoustmultivariablehypergeometricfunction.*)
In[]:=
H1
1
10
,
1
100
,20
Out[]=
0.449737
See Here for definitionfor: Srivastava - Daoust multivariable hypergeometric function .