Question: How to use Gauss multiplication formula to simplify the two products?
In[]:=
p2[x_,n_]:=NProduct
-
1-p
2
(2π)
-p
2
-
3
2
+4p+
-p
5
x
5
Gamma[-1+
-1-p
5
x]Gamma
1
10
+
-1-p
5
xGamma
3
10
+
-1-p
5
xGamma
7
10
+
-1-p
5
xGamma
9
10
+
-1-p
5
x
Gamma
1
2
+
-1-p
5
x
Gamma[1+
-1-p
5
x]
(
p
5
-x)(2
p
5
-x)(3
p
5
-x)(4
p
5
-x)Gamma[-5+
-p
5
x]
,{p,0,n},
1-
-1-n
2
5
Gamma
1
2
+x
-1-n
2
Gamma[1+
-1-n
5
x]
Gamma
1
2
+
-1-n
5
x
Gamma[1+x]
//N
In[]:=
p2[9+,3]
Out[]=
{1.42971-0.0768096,1.42971-0.0768096}
In[]:=
p2[9,4]
Out[]=
{1.53555,1.53555}
In[]:=
p2[π,3]
Out[]=
{2.36675,2.36675}
In[]:=
p2[π,6]
Out[]=
{2.66553,2.66553}
In[]:=
p3[x_]:=NProduct
-
1-p
2
(2π)
-p
2
-
3
2
+4p+
-p
5
x
5
Gamma[-1+
-1-p
5
x]Gamma
1
10
+
-1-p
5
xGamma
3
10
+
-1-p
5
xGamma
7
10
+
-1-p
5
xGamma
9
10
+
-1-p
5
x
Gamma
1
2
+
-1-p
5
x
Gamma[1+
-1-p
5
x]
(
p
5
-x)(2
p
5
-x)(3
p
5
-x)(4
p
5
-x)Gamma[-5+
-p
5
x]
,{p,0,∞},VerifyConvergenceFalse,Method"WynnEpsilon",
5Gamma
1
2
+x
Gamma[1+x]
//N
In[]:=
p3[3+]
Out[]=
{2.67866-0.400748,2.67866-0.400748}