(1/(2^j)((k*Gamma[5+2j]Gamma[1+l]HypergeometricPFQ[{1,5/2+j,3+j},{3+j+l/2,7/2+j+l/2},z])/Gamma[6+2j+l]+((k+m)Gamma[7+2j]Gamma[1+l]HypergeometricPFQ[{1,7/2+j,4+j},{4+j+l/2,9/2+j+l/2},z])/Gamma[8+2j+l]))/(2^(-5-3j-l)Gamma[5+2j]Gamma[1+l](kHypergeometricPFQRegularized[{1,5/2+j,3+j},{3+j+l/2,7/2+j+l/2},z]+1/2(3+j)(5+2j)(k+m)HypergeometricPFQRegularized[{1,7/2+j,4+j},{4+j+l/2,9/2+j+l/2},z]))
a/bwithaCtrl/bbecomes
a
b
​​2^jwith2Ctrl6jbecomes
j
2
1
j
2
k*Gamma[5+2j]Gamma[1+l]
Gamma[6+2j+l]
HypergeometricPFQ[{1,5/2+j,3+j},{3+j+l/2,7/2+j+l/2},z]+
1
j
2
(k+m)Gamma[7+2j]Gamma[1+l]
Gamma[8+2j+l]
HypergeometricPFQ[{1,7/2+j,4+j},{4+j+l/2,9/2+j+l/2},z]
-5-3j-l
2
Gamma[5+2j]Gamma[1+l]kHypergeometricPFQRegularized[{1,5/2+j,3+j},{3+j+l/2,7/2+j+l/2},z]+
(3+j)
2
(5+2j)(k+m)HypergeometricPFQRegularized[{1,7/2+j,4+j},{4+j+l/2,9/2+j+l/2},z]
​
HypergeometricPFQRegularized[{a,b,c},{e,f},z]=
1
Gamma[e]Gamma[f]
HypergeometricPFQ[{a,b,c},{e,f},z]​​Thisisthedefinition.SeetheHelpfunction​​
1
j
2
k*Gamma[5+2j]Gamma[1+l]
Gamma[6+2j+l]
-5-3j-l
2
Gamma[5+2j]Gamma[1+l]
HypergeometricPFQ[{1,5/2+j,3+j},{3+j+l/2,7/2+j+l/2},z]+
1
j
2
(k+m)Gamma[7+2j]Gamma[1+l]
Gamma[8+2j+l]
-5-3j-l
2
Gamma[5+2j]Gamma[1+l]
HypergeometricPFQ[{1,7/2+j,4+j},{4+j+l/2,9/2+j+l/2},z]
k
Gamma3+j+
l
2
Gamma72+j+
l
2

HypergeometricPFQ[{1,5/2+j,3+j},{3+j+l/2,7/2+j+l/2},z]+
(3+j)(5+2j)(k+m)
2Gamma4+j+
l
2
Gamma92+j+
l
2

HypergeometricPFQRegularized[{1,7/2+j,4+j},{4+j+l/2,9/2+j+l/2},z]
Gamma[2a]=
1
π
Gamma[a]Gammaa+
1
2

2a-1
2
​​ThisistheduplicationformulafortheGammafunction.Seehttps://dlmf.nist.gov/5.5Formula5.5​​
5+2j+l
2
k
Gamma[6+2j+l]
HypergeometricPFQ1,
5
2
+j,3+j,3+j+
l
2
,
7
2
+j+
l
2
,z+
5+2j+l
2
(k+m)Gamma[7+2j]
Gamma[8+2j+l]Gamma[5+2j]
HypergeometricPFQ1,
7
2
+j,4+j,4+j+
l
2
,
9
2
+j+
l
2
,z
k
5+2j+l
2
π
Gamma[6+2j+l]
HypergeometricPFQ1,
5
2
+j,3+j,3+j+
l
2
,
7
2
+j+
l
2
,z+
(3+j)(5+2j)(k+m)
6+2j+l
2
π
Gamma[8+2j+l]
HypergeometricPFQ1,
7
2
+j,4+j,4+j+
l
2
,
9
2
+j+
l
2
,z
Gamma[a+1]=aGamma[a]ThisfollowsfromthedefinitionoftheGammafunction.Seehttps://dlmf.nist.gov/5.4.1​​ApplicationgivesGamma[7+2j]=(6+2j)Gamma[6+2j]=(6+2j)(5+2j)Gamma[5+2j]​​(3+j)(5+2j)(k+m)
6+2j+l
2
=(6+2j)(5+2j)(k+m)
5+2j+l
2
π
k
Gamma[6+2j+l]
HypergeometricPFQ1,
5
2
+j,3+j,3+j+
l
2
,
7
2
+j+
l
2
,z+
(k+m)(6+2j)(5+2j)
Gamma[8+2j+l]
HypergeometricPFQ1,
7
2
+j,4+j,4+j+
l
2
,
9
2
+j+
l
2
,z
k
Gamma[6+2j+l]
HypergeometricPFQ1,
5
2
+j,3+j,3+j+
l
2
,
7
2
+j+
l
2
,z+
(k+m)(6+2j)(5+2j)
Gamma[8+2j+l]
HypergeometricPFQ1,
7
2
+j,4+j,4+j+
l
2
,
9
2
+j+
l
2
,z