Using Christian Krattenthaler's hyp.m tells you the following - the important information is at the end, and the result agrees with Johannes' comment.
In[1]:= <<hyp.m
Out[1]= ▒
In[10]:= S = SUM[1/(n+1/2)*(Gamma[n]^2/Gamma[n+1/2]^2),{n,1,Infinity}]
Infinity
----- 2
\ Gamma[n]
Out[10]= > ---------------------
/ 1 1 2
----- (- + n) Gamma[- + n]
n=1 2 2
In[19]:= SF = S/.SUMF
[ 1, 1, 1 ]
| | 2
2 F | 5 3 ; 1 | Ga(1)
3 2| -, - |
[ 2 2 ]
Out[19]= --------------------------
3 2
3 Ga(-)
2
In[21]:= SF/.SListe
Be sure to apply "FOrdne" before using the following information!
2
2 S3261 Ga(1)
Out[21]= {{--------------}}
3 2
3 Ga(-)
2
In[22]:= SF/.S3261
1 1 1 3 3 1 1 1 3
[ -(-), -(-), -(-), - ] [ 1 1 ] [ -, -(-), -(-), -(-), - ]
2 | 2 2 2 2 | | -(-), -(-) | | 2 2 2 2 2 |
2 Ga(1) (Ga| | - F | 2 2 ; 1 | Ga| |)
| 3 1 1 1 | 2 1| | | 3 |
[ -(-), -, -, - ] [ 0 ] [ -(-), 1, 1, 1, 0 ]
2 2 2 2 2
Out[22]= ----------------------------------------------------------------------------------------
3 2
3 Ga(-)
2
In[23]:= SF/.S3261/.SListe
Be sure to apply "FOrdne" before using the following information!
1 1 1 3 3 1 1 1 3
[ -(-), -(-), -(-), - ] [ -, -(-), -(-), -(-), - ]
2 | 2 2 2 2 | | 2 2 2 2 2 |
2 Ga(1) (Ga| | - S2103 Ga| |)
| 3 1 1 1 | | 3 |
[ -(-), -, -, - ] [ -(-), 1, 1, 1, 0 ]
2 2 2 2 2
Out[23]= {{-------------------------------------------------------------------------}}
3 2
3 Ga(-)
2
In[33]:= R = SF/.S3261/.S2103/.Gzerl
1 3 3 1 3 3 2
Ga(-(-)) Ga(-) Ga(-(-)) Ga(-)
2 2 2 2 2
2 Ga(1) (--------------- - ----------------------)
3 1 3 3 1 2 2
Ga(-(-)) Ga(-) Ga(-(-)) Ga(-) Ga(1)
2 2 2 2
Out[33]= ---------------------------------------------------
3 2
3 Ga(-)
2
In[42]:= Simplify[R]
1 3 2 1 3
2 Ga(-(-)) (Ga(1) - Ga(-) Ga(-))
2 2 2
Out[42]= ----------------------------------
3 1 3 3
3 Ga(-(-)) Ga(-) Ga(-)
2 2 2
In[52]:= ?S3261
Summation formula (Slater, Appendix (III.31)) in form of a rule.
In[53]:= ?S2103
Summation formula (Slater, Appendix (III.3)) in form of a rule.