
 
%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\Tilde{\char126\relax}
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
 0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
 height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
%end macros
 

\bf
\centerline
{
52 Strange Evaluations by Shalosh B. Ekhad
}


Theorem 1 : 
$$
F(-n \, , \, -4n-1/2 \, , \, -3n \, , \, -1 )\,=\,(64/27)^n \, \cdot \,
{{(3/8)_n(5/8)_n} \over {(1/3)_n(2/3)_n}} \quad .
$$

Proof : 
$$
{
{-2(296n^3-216n^2k+771n^2+651n-372nk+52nk^2-155k+177-4k^3+45k^2)(3n-k+1)k}
\over
{(8n+9-2k)(8n+7-2k)(8n+5-2k)(8n+3-2k)(n-k+1)}} \quad 
\halmos
$$

Theorem 2 : 
$$
F(-n \, , \, -4n-5/2 \, , \, -3n-1 \, , \, -1 )\,=\,
(64/27)^n \, \cdot \,
{{(7/8)_n(9/8)_n} \over {(2/3)_n(4/3)_n}} \quad.
$$
Proof : 
$$
(296n^3+1215n^2-216n^2k+1660n+52nk^2-
586nk+756-394k-4k^3+71k^2)
\cdot
$$
$$
{{
-2
(3n-k+2)k} \over
{(8n+13-2k)(8n+11-2k)(8n+9-2k)(8n+7-2k)(n-k+1)}}  \quad \halmos
$$

Theorem 3 : 
$$
F(-n \, , \, -3n-1/2 \, , \, -4n \, , \, 4 )\,=\,
(-27/16)^n \, \cdot \,
{{(1/3)_n(5/6)_n} \over {(1/4)_n/(3/4)_n} } \quad .
$$

Proof : 
$$
{{(344n^3-216n^2k+768n^2+522n-282nk+34nk^2-75k+104+13k^2)
(4n-k+1)k} \over
{4 (6n+7-2k)(6n+5-2k)(6n+3-2k)(n-k+1)(3n+1)}}  \quad \halmos.
$$

Theorem 4 : 
$$
F(-n \, , \, -3n-3/4 \, , \, -4n-1 \, , \, -8 )\,=\,
(27/4)^n \, \cdot \,
{{(2/3)_n(7/12)_n} \over {(1/2)_n(3/4)_n}} .
$$

Proof : 
$$
{
{-(2208n^3-1488n^2k+6272n^2+5762n-2620nk+252nk^2-1082k+1707+167k^2)
(4n-k+2)k}
\over
{
12(12n+15-4k)(12n+11-4k)(12n+7-4k)(n-k+1/(3n+2)}}  \quad \halmos
$$

Theorem 5 : 
$$
F(-n \, , \, -3n-3/2 \, , \, -4n-2 \, , \, -8 )\,=\,(27/4)^n \, \cdot \,
{
{(5/6)_n(7/6)_n}
\over
{(3/4)_n(5/4)_n}} \quad.
$$

Proof : 
$$
-(1/24)
{{(184n^2+492n-124nk+326+21k^2-167k)(4n-k+3)k}
\over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n-k+1)}}  \quad \halmos
$$


Theorem 6 : 
$$
F(-n \, , \, -3n-1/4 \, , \, -4n  \, , \, -8 )\,=\,
(27/4)^n \, \cdot \,
{{(1/3)_n(5/12)_n} \over {(1/2)_n/(1/4)_n}} .
$$

Proof : 
$$
{{(-1/4)(736n^3-496n^2k+1600n^2+1058n+84nk^2-624nk-149k+200+27k^2)(4n-k+1)k}
\over
{(12n+13-4k)(12n+9-4k)(12n+5-4k)(n-k+1)(3n+1)}}  \quad \halmos
$$

Theorem 7 : 
$$
F(-n \, , \, -3n-1/2 \, , \, -4n \, , \, -8 )\,=\,
(27/4)^n \, \cdot \,
{{(1/3)_n(2/3)_n} \over
{ (1/4)_n(3/4)_n}} .
$$
Proof : 
$$
(-1/32)(2208n^4-1488n^3k+6640n^3+252n^2k^2-2988n^2k+7104n^2+248nk^2-1812nk+
3156n+53k^2-321k+478) \cdot
$$
$$
{{
(4n-k+1)k} \over
{(6n+7-2k)(6n+5-2k)(6n+3-2k)(n-k+1)(3n+2)(3n+1)}}  \quad \halmos
$$

Theorem 8 : 
$$
F(-n \, ,\,  -3n-5/4 \, , \, -4n-1 \, , \, -8 )\,=\,
(27/4)^n \, \cdot \,
{{(2/3)_n(13/12)_n} \over {(1/2)_n(5/4)_n}} .
$$
Proof : 
$$
{
{(-1/4)(736n^3-496n^2k+2336n^2+2382n-956nk+84nk^2-414k+767+55k^2)(4n-k+2)k}
\over
{(12n+17-4k)(12n+13-4k)(12n+9-4k)(n-k+1)(3n+2)}}  \quad \halmos
$$

Theorem 9 : 
$$
F(-n \, , \, -3n-3/2 \, , \, -4n-2 \, , \, 4 )\,=\,
(-27/16)^n \, \cdot \,
{{(4/3)_n(5/6)_n} \over {(3/4)_n(5/4)_n }}.
$$

Proof : 
$$
{{
(1/12)(1032n^3-648n^2k+4024n^2+5202n-1690nk+102nk^2-1103k+2228+135k^2)
(4n-k+3)k} \over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n-k+1)(3n+4)}}  \quad \halmos
$$

Theorem 10 : 
$$
F(-n \, , \, -3n-7/4 \, , \, -4n-2 \, , \, -8 ) \, = \,
(27/4)^n \, \cdot \,
{{(4/3)_n(11/12)_n} \over {(3/2)_n(3/4)_n}} .
$$
Proof : 
$$
{{(-1/12)(2208n^3+9216n^2-1488n^2k+12806n+252nk^2-4112nk+5924-2839k+335k^2)
(4 n-k+3)k} \over {
(12n+19-4k)(12n+15-4k)(12n+11-4k)(n-k+1)(3n+4)}}  \quad \halmos
$$

Theorem 11 : Let $\omega$ be a primitive cubic root of unity
(i.e. $\omega\,=\,(1 \pm \sqrt{3} i)/2$), then
$$
F(-n \, , \, -3n-1 \, , \, -2n \, , \, \omega )\,=\,
(-(3/2) \omega+3/4)^n \, \cdot \,
{{(2/3)_n)} \over {(1/2)_n }} \quad .
$$
Proof : 
$$
(129n^2+289n-2\omega n+160-2\omega - 88kn+14k\omega n-
101k+19k \omega +13k^2-5 \omega k^2)
\cdot
$$
$$
{{(-1/129)
(-5+13\omega)
(2n-k+1)k} \over
{
(3n+4-k)(3n+3-k)(3n+2-k)(n-k+1) }}
\quad  \halmos
$$


Theorem 12 :
$$
F(-n \, , \, -3n-1/2 \, , \, -n+1/2 \, , \,4 )\, = \, (-27)^n .
$$

Proof : 
$$
{{
(1/9)k(2n+1-2k)(84n^2+172n-60nk+12k^2-64k+87)}
\over
{(6n+7-2k)(6n+5-2k)(6n+3-2k)(n+1-k)}} \quad  \halmos
$$


Theorem 13 : 
$$
F(-n \, , \, -3n-3/2 \, , \, -n+1/2 \, , \, 4 ) \, = \,
(-27)^n \, \cdot \,
{{(5/6)_n(7/6)_n}
\over
{(1/2)_n(3/2)_n}} \quad .
$$

Proof : 
$$
{{
(1/3)(28n^2+76n-20nk+51-28k+4k^2)(2n+1-2k)k}
\over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n+1-k)}}  \quad \halmos
$$

Theorem 14 :  Let $\alpha \, = \,9 \pm 4 \sqrt{5}$, then
$$
F(-n \, , \, -3n-1/2  \, , \, 2n+3/2 \, ,  \,
\alpha  
)
\, = \,
(128/125+128/125\alpha )^n \, \cdot \,
{{(3/4)_n(5/4)_n} \over
{(4/5)_n (6/5)_n}}\quad .
$$

Proof : 
$$
(79794+388827n+706989n^2+566136n^3+167796n^4-4518k-4875nk-51972k^2+11429
\alpha nk+
$$
$$
3754\alpha k+2132\alpha k^2+496k^4\alpha +
26472k^3-3696k^4-6816kn^3-6174kn^2-127434k^2n-72852k^2n^2+30768k^3n
$$
$$
-3336n^3\alpha -9466\alpha n^2-8839\alpha n-2718\alpha +10998k\alpha 
n^2+1060k^2\alpha n^2-2192k^3\alpha n+3336kn^3\alpha +3058k^2\alpha n-2584k^3
\alpha ) \cdot
$$
$$
{{
-(1/74576)(31 \alpha -327) k} \over
{
(6n+7-2k)(6n+5-2k)(6n+3-2k)(n-k+1)(4n+3+2k)
(\alpha -1) 
}} .
\quad \halmos
$$

Theorem 15 : Let $\alpha \, = \, 9 \pm 4 \sqrt{5}$, then
$$
F(-n \, , \, -3n-3/2 \, , \, 2n+5/2 \, , \, \alpha  ) \, = \,
(128/125+128/125\alpha )^n \, \cdot \,
{{(5/4)_n(7/4)_n}
\over
{(7/5)_n(8/5)_n}} \quad .
$$

Proof : 
$$
(384115+1282296n+1575413n^2+845796n^3+167796n^4-86193k-
153105nk-77610k^2+26963\alpha nk+
$$
$$
13383\alpha k+730\alpha k^2+
496k^4\alpha +34264k^3-3696k^4-6816kn^3-
$$
$$
74602kn^2-154458k^2n-
72852k^2n^2+30768k^3n-3336n^3\alpha -12126\alpha n^2-14517\alpha n-
$$
$$
5745\alpha +17114k\alpha n^2+1060k^2\alpha n^2-2192k^3\alpha n+
3336kn^3\alpha +1842k^2\alpha n-2984k^3\alpha ) \cdot
$$
$$
{{
k(-1/74576)
(31\alpha -327)
}
\over
{(6n+9-2k)(6n+7-2k)(6n+5-2k)(n-k+1)(4n+5+2k)(\alpha -1)}}  \quad \halmos
$$

Theorem 16 : Let $\alpha \, =2 \, \pm \sqrt{3}$, then
$$
F(-n \, , \, -2n-1/2 \, , \, -4n-1 \, , \,4\alpha  ) \, = \,
(3/4-3/2\alpha )^n \, \cdot \,
{{(2/3)_n} \over {(1/2)_n}} \quad .
$$

Proof : 
$$
(488n^2+978n-20\alpha n+472-23\alpha +94k^2-
25k^2\alpha -420kn+52kn\alpha -422k+72k\alpha )
\cdot
$$
$$
{{
(1/732)
(-68+19\alpha )(4n+2-k)k} \over {
(4n+5-2k)(4n+3-2k)(n-k+1)(3n+2)}}  \quad \halmos
$$

Theorem 17 : 
$$
F(-n \, , \, -2n-1/3 \, , \, -3n \, , \, 9 )\, = \,
(-4)^n \, \cdot \,
{{(1/2)_n} \over {(1/3)_n}} \quad .
$$

Proof : 
$$
{{
(1/9)k(3n-k+1)(45n^2-21nk+72n-11k+25)} \over
{(6n+7-3k)(6n+4-3k)(n-k+1)(2n+1)}}  \quad \halmos.
$$

Theorem 18 : 
$$
F(-n \, , \, -2n-2/3 \, ,\, -3n-1 \,, \,9 ) \, = \,
(-4)^n \, \cdot \,
{{(5/6)_n} \over {(2/3)_n}} \quad .
$$

Proof : 
$$
{{
(1/6)(15n+19-7k)(3n-k+2)k} \over
{
(6n+8-3k)(6n+5-3k)(n-k+1)}}  \quad \halmos
$$

Theorem 19 : 
$$
F(-n \, , \, -n-1/2 \, , \, 4n+9/2 \, , \, 1/5 )
\, = \,
({ {16384} \over {15625}} )^n \, \cdot \,
{{
(9/8)_n(11/8)_n(13/8)_n(15/8)_n } \over
{(6/5)_n(9/5)_n(13/10)_n(17/10)_n}} \quad.
$$

Proof : 
$$
(3036n^4+1256n^3k+16960n^3+35275n^2+4780n^2k-1184n^2k^2+6180nk-
$$
$$
3440nk^2-544nk^3+32445n+2720k+11154-2480k^2-64k^4-800k^3)
\cdot
$$
$$
{{
(-5/16)
k}
\over
{
(2n+3-2k)(n-k+1)(8n+13+2k)(8n+11+2k)(8n+9+2k)}}  \quad \halmos
$$

Theorem 20 : 
$$
F(-n \, , \, -n+1/2 \, , \, 4n+3/2 \, , \, 1/5 ) \, = \,
( {{16384} \over {15625} })^n \, \cdot \,
{{(3/8)_n(7/8)_n(5/8)_n(9/8)_n}
\over
{(3/5)_n(2/5)_n(9/10)_n(11/10)_n}} \quad .
$$

Proof : 
$$
(3036n^4+1256n^3k+7852n^3+7409n^2+2844n^2k-1184n^2k^2+2252nk-1344nk^2+2993n-
$$
$$
544nk^3-336k^2+592k+430-352k^3-64k^4)
\cdot
$$
$$
{{
(-5/16)
k
} \over {
(2n+1-2k)(n-k+1)(8n+7+2k)(8n+5+2k)(8n+3+2k)}}  \quad \halmos
$$

Theorem 21 :  Let $\beta \, = \, (3 \pm sqrt{5})/2$, then
$$
F(-n \, , \, 2n+1 \, , \, -3n \, , \, \beta  )
\, = \,
((25/27)+(25/27)\beta )^n \, \cdot \,
{
{(2/5)_n(3/5)_n} \over
{(1/3)_n(2/3)_n}} \quad.
$$

Proof : 
$$
(1376n+308+2026n^2+958n^3-60k-23k^3\beta +38k^2\beta -
91k\beta +139\beta n+87\beta n^2
$$
$$
-201k\beta n-
87k\beta n^2+38k^2\beta n+48k^3+4k^2-122kn-110kn^2+4k^2n+52\beta )
\cdot
$$
$$
{{
(1/4790)
k(3n-k+1)(-78+17\beta )
} \over {
(n-k+1)(5n+3)(5n+2)(n+1)(2n+1)}}  \quad \halmos
$$

Theorem 22 : 
Let $\beta \, = \, (3 \pm sqrt{5})/2$, then
$$
F(-n \, , \, 2n+2 \, , \, -3n-1 \, , \, \beta  ) \, = \,
(25/27+25/27\beta )^n \, \cdot \,
{{(4/5)_n(6/5)_n} \over
{(2/3)_n(4/3)_n}} \quad.
$$
Proof : 
$$
(4179n+3463n^2+958n^3+24k-317k\beta n-87k\beta n^2+48k^3+54k^2-
109kn-110kn^2+4k^2n+96\beta +
$$
$$
1674+183\beta n+
87\beta n^2+38k^2\beta n-23k^3\beta +34k^2\beta -251k\beta )
\cdot
$$
$$
{{
(1/4790)
k(3n-k+2)(-78+17\beta )
}
\over {
(n-k+1)(5n+6)(5n+4)(2n+3)(n+1)}}  \quad \halmos
$$

Theorem 23 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$ then
$$
F(-n \, , \, 2n+1 \, , \, 4n+2 \, , \,\phi^{-1}  )
\, = \,
(128/125-64/125\phi^{-1} )^n \, \cdot \,
{{
(3/4)_n(5/4)_n} \over
{(4/5)_n (6/5)_n}} \quad.
$$

Proof : 
$$
(2815n+3661n^2+1558n^3+14k^3\phi^{-1} -
148\phi^{-1} n-84\phi^{-1} n^2+579k+183k\phi^{-1} n+
$$
$$
1567kn+1013kn^2+88k\phi^{-1} +37k^2\phi^{-1} n+34k^2\phi^{-1} +
$$
$$
270k^2n+84k\phi^{-1} n^2+712+206k^2+39k^3-64\phi^{-1} )
\cdot
$$
$$
{{
(-1/779)k
(14\phi^{-1} -25)} \over
{(n-k+1)(4n+4+k)(4n+3+k)(4n+2+k)(-1+\phi^{-1} )}}  \quad \halmos
$$

Theorem 24 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$ then
$$
F(-n \, , \,  2n+2 \, , \, 4n+4 \, ,  \, \phi^{-1}  ) \, = \,
(128/125-64/125\phi^{-1} )^n \, \cdot \,
{{(5/4)_n(7/4)_n} \over
{(7/5)_n(8/5)_n}} \quad.
$$

Proof : 
$$
(7628n+5998n^2+1558n^3+14k^3\phi^{-1} -
178\phi^{-1} n
$$
$$
-84\phi^{-1} n^2+3188+1762k+ 309k\phi^{-1} n+2697kn+1013kn^2+253k\phi^{-1} +
$$
$$
37k^2\phi^{-1} n+79k^2\phi^{-1} +270k^2n+84k\phi^{-1} 
n^2+387k^2+39k^3-94\phi^{-1} )
\cdot
$$
$$
{{
(-1/779)
k
(14\phi^{-1} -25)} \over {
(n-k+1)(4n+6+k)(4n+5+k)(4n+4+k)(-1+\phi^{-1} )}}  \quad \halmos
$$

Theorem 25 : Let $\gamma \, = \, 3 \pm 2 \sqrt{2}$, then
$$
F(-2n \, , \, -4n-1/2 \, , \, -2n+1/2 \, , \, \gamma  ) \, = \,
(-16+96\gamma )^n
{{
(3/8)_n(5/8)_n} \over
{(1/4)_n(3/4)_n}} \quad.
$$
Proof : 
$$
(25434+137162n+271600n^2+233440n^3+73472n^4+5216k^2\gamma n^2+
10144k^2\gamma n-21148k\gamma n-
$$
$$
6208k\gamma n^3-20304k\gamma n^2-1264k^3\gamma n-
142224kn^2-58944kn^3-111324kn+14304k^2n^2+
$$
$$
22536k^2n-1776k^3n-1372k^3+8594k^2+88k^4+4834k^2\gamma -
7067k\gamma -1204k^3\gamma -28145k+3837\gamma +
$$
$$
104k^4\gamma +13852\gamma n+6208\gamma n^3+16240\gamma n^2)
\cdot
$$
$$
{{
(-1/1148)(-89+13\gamma )
(4n+1-2k)k} \over {
(8n+9-2k)(8n+7-2k)(8n+5-2k)(8n+3-2k)(2n-k+1)(2n-k+2)(\gamma -1)}}  
\quad \halmos
$$
Theorem 26 : 
$$
F(-2n \, , \, -4n-1/3 \, , \, 2/3 \, , \, -8 ) \, =729^n \, .
$$

Proof : 
$$
(314496n^4+1101600n^3+1416600n^2+789644n+
160228-1012068kn^2-393984kn^3-
$$
$$
846054kn-229356k+172044k^2n^2+
290142k^2n+119151k^2-32076k^3n-26730k^3+2187k^4)
\cdot
$$
$$
{{
(1/729)
(-1+3k)k
} \over {
(2n-k+1)(2n-k+2)(12n+4-3k)(12n+7-3k)(12n+10-3k)(12n+13-3k)}}  
\quad \halmos
$$

Theorem 27 : Let $\delta \, = \, 5 \pm 3 \sqrt{3}$, then
$$
F(-2n \, , \, -4n-1/3 \, , \, 2/3 \, , \, 2\delta  ) \, = \,
(702\delta +135)^n \, \cdot \,
{{(7/12)_n} \over {(3/4)_n}}
 \quad .
$$
Proof : 
$$
(2059512+12099448n+766292\delta n+
25671232n^2+23410464n^3+7767936n^4+933000\delta n^2-
$$
$$
1239770k\delta n-404352kn^3\delta +506562k^2\delta 
n+268020k^2\delta n^2-62100k^3\delta n-1257180k\delta 
n^2
$$
$$
-2121648k-5589216kn^3-12542136kn^2-9090980kn+1684440k^2n^2+2508684k^2n
$$
$$
-240624k^3n-56934k^3\delta +232953k^2\delta 
-390660k\delta +370656n^3\delta 
$$
$$
+205020\delta +13446k^4-178812k^3+904542k^2+4941k^4\delta )
\cdot
$$
$$
{{
(1/364122)k(-1+3k)
(61\delta -776)} \over {
(12n+13-3k)(12n+10-3k)(12n+7-3k)(12n+4-3k)(2n-k+1)(2n-k+2)
2\delta -1)}}  \quad \halmos
$$

Theorem 28 : Let $\delta \, = \, 5 \pm 3 \sqrt{3}$, then
$$
F(-2n \, , \, -4n-2/3 \, , \, 4/3 \, , \, 2\delta  )=
(702\delta +135)^n \, \cdot \,
{{(5/12)_n} \over {(5/4)_n}} \quad.
$$
Proof : 
$$
(3710744+18504312n+607916\delta n+
33746032n^2+26726016n^3+7767936n^4+661656\delta n^2-
$$
$$
1315958k\delta n-404352kn^3\delta +533802k^2\delta n+
268020k^2\delta n^2-62100k^3\delta n-1280676k\delta n^2-
$$
$$
3169096k-5589216kn^3-14213784kn^2-11770892kn+1684440k^2n^2+
2829540k^2n-240624k^3n
$$
$$
-60696k^3\delta +260835k^2\delta -
439228k\delta +235872n^3\delta +182668\delta 
$$
$$
+13446k^4-200988k^3+1159494k^2+4941k^4\delta )
\cdot
$$
$$
{{
(1/364122)k(1+3k)
(61\delta -776)
} \over {
(12n+14-3k)(12n+11-3k)(12n+8-3k)(12n+5-3k)(2n-k+1)(2n-k+2)(2\delta -1)
}}  \quad \halmos 
$$

Theorem 29 : 
$$
F(-2n \, , \, -4n-2/3 \, , \, 4/3 \, , \, -8 ) \, = \, 
729^n \, \cdot \,
{{(5/12)_n(11/12)_n}\over
{(3/4)_n(5/4)_n}} \quad.
$$

Proof : 
$$
(34944n^4-43776kn^3+131104n^3-121596kn^2+181768n^2+19116k^2n^2-
$$
$$
110586kn+110164n+35046k^2n-3564k^3n+15741k^2+243k^4-32868k+24564-3240k^3)
\cdot
$$
$$
{{
(1/81)
(1+3k)k} \over
{
(12n+14-3k)(12n+11-3k)(12n+8-3k)(12n+5-3k)(2n-k+1)(2n-k+2)}}  
\quad \halmos
$$

Theorem 30 : Let $\alpha \, = \, 2 \pm \sqrt{3}$, then
$$
F(-2n \, , \, -3n-1/4 \, , \, -4n \, , \, 4\alpha  ) \, = \, 
(39/2\alpha -21/4)^n \, \cdot \,
{{(5/12)_n} \over {(1/4)_n}} \quad.
$$

Proof : 
$$
(171978n+238000n^2+104736n^3-12496k\alpha n^2-
32440k\alpha n+7780k^2\alpha n+17488\alpha n^2+
$$
$$
30900\alpha n+8407k^2\alpha -1216k^3\alpha -18581k\alpha +
39182+13262\alpha 
$$
$$
-81808kn^2+15490k^2-1792k^3-43403k-
122224kn+21112k^2n)
\cdot
$$
$$
{{
(-1/3273)
k(4n-k+1)
(397\alpha -1484)
} \over
{
(12n+13-4k)(12n+9-4k)(12n+5-4k)(2n-k+1)(2n-k+2)}}  
\quad \halmos
$$

Theorem 31 : 
$$
F(-2n \, , \, -3n+1 \, , \, -2n+2 \, , \, -2 )\, = \, 27^n \, \cdot \,
{{(1/2)_n} \over {(-1/2)_n}} \quad .
$$

Proof : 
$$
{{
(-1/9)
k(2n-k-1)
(39n^2+55n-21kn-14k+3k^2+18)
} \over {
(3n+2-k)(3n+1-k)(3n-k)(2n+2-k)}}  \quad \halmos
$$

Theorem 32 : 
$$
F(-2n \, , \, -3n \, , \,  2n+1/2 \, , \, 1/4 ) \, = \, 
(27/64)^n \, \cdot \,
{{(1/3)_n (2/3)_n} \over
{(1/4)_n(3/4)_n }} \quad.
$$

Proof :
$$
{{
( -2/3)(74n^3+168n^2-57kn^2-87kn+124n+12k^2n-32k+30-k^3+9k^2)(4n+1-2k)k
} \over {
(3n+3-k)(3n+2-k)(3n+1-k)(2n+1-k)(2n+2-k)}}  \quad \halmos
$$

Theorem 33 : Let $\eta \, = \, -7 \pm 4 \sqrt{3}$, then
$$
F(-2n \, , \,  -3n-1/4 \, , \, -n+3/4 \, , \, \eta  ) \, = \, 
(312\eta +24)^n \, \cdot \,
{{(5/12)_n} \over {(1/4)_n}} \quad .
$$

Proof : 
$$
(485694n+611008n^2+248736n^3-5600k^2\eta n+
11504k\eta n^2+23720\eta nk-11880\eta n-
$$
$$
6512\eta n^2+752k^3\eta +22744k^2+11851\eta k-5432\eta k^2-
204544nk-79751k-2416k^3
$$
$$
+28576k^2n-126928kn^2-5299\eta +124358)
\cdot
$$
$$
{{
(1/124368)
(47\eta +809)
(4n+1-4k)k
} \over
{(12n+13-4k)(12n+9-4k)(12n+5-4k)(2n-k+1)(2n-k+2)(\eta -1)}}  \quad \halmos
$$

Theorem 34 : 
$$
F(-2n \, , \, -2n+1/3 \, , \, -n+5/6 \, , \, -1/8  \, )= \,
(27/16)^n \, \cdot \,
{{(1/3)_n} \over {(1/6)_n}} \quad .
$$

Proof : 
$$
{{
(-4/27)(132n^2-60nk+202n+74+9k^2-45k)(6n+1-6k)k}
\over
{(6n+5-3k)(6n+2-3k)(2n-k+1)(2n-k+2)}}  \quad \halmos
$$

Theorem 35 : 
$$
F(-2n \, , \, -2n+1/2 \, , \, 4n+3/2 \, , \,  -1/3 ) \, = \,
(4096/6561)^n \, \cdot \,
{{(3/8)_n(7/8)_n(5/8)_n(9/8)_n} \over
{(1/3)_n/(5/6)_n/(7/12)_n/(13/12)_n}} \quad.
$$

Proof : 
$$
(21483+241989n+1106816n^2+2625708n^3+3409136n^4+2299968n^5+631040n^6-
$$
$$
405096kn^2-723616kn^3-213504kn^5-628608kn^4-161808k^2n^2-175872k^2n^3-
72960k^2n^4-10936k^2
$$
$$
-109724kn+31744k^3n^3+5856k^3-
67368k^2n-768k^5+5376k^4n^2+6720k^4n+2624k^4
$$
$$
+31136k^3n+54144k^3n^2-
256k^6-1536k^5n-11388k)
\cdot
$$
$$
{{
(-3/32)
k
} \over {
(4n+3-2k)(4n+1-2k)(2n-k+2)(2n-k+1)(8n+7+2k)(8n+5+2k)(8n+3+2k)}}  
\quad \halmos
$$

Theorem 36 : 
$$
F(-2n \, , \, -2n+1/4 \, , \, 4n+5/4 \, , \, 1/9 ) \, = \,
(65536/59049)^n \, \cdot \,
{{(9/16)_n(5/16)_n(13/16)_n(17/16)_n} \over
{(1/3)_n(5/6)_n(13/24)_n(25/24)_n}} \quad.
$$

Proof : 
$$
(501813+5630279n+25382756n^2+59064356n^3+75057056n^4+49523456n^5+
13285376n^6+1953360kn^2
$$
$$
-59200kn^3-1363968kn^5-
2306048kn^4-9668544k^2n^2-9942016k^2n^3-3846144k^2n^4
$$
$$
-684064k^2+
1280440kn+753664k^3n^3+36608k^3-4189920k^2n
$$
$$
-16384k^5+368640k^4n^2+523264k^4n+
195584k^4+446720k^3n+1058816k^3n^2-16384k^6-49152k^5n+247832k)
\cdot
$$
$$
{{
(-9/128)
k
} \over {
(8n+7-4k)(8n+3-4k)(2n-k+2)(2n-k+1)(16n+13+4k)(16n+9+4k)(16n+5+4k)}}  
\quad \halmos
$$


Theorem 37 : 
$$
F(-2n \, , \, -2n+1/2 \, , \, 4n+3/2 \, , \, 1/9 ) \, = \,
(65536/59049)^n \, \cdot \,
{{(3/8)_n(7/8)_n(5/8)_n(9/8)_n} \over {
(7/12)_n(5/12)_n(11/12)_n(13/12)_n}} \quad.
$$

Proof : 
$$
(693315+7099938n+29906648n^2+66047424n^3+80454656n^4+51184128n^5+13285376n^6
$$
$$
-3206400kn^2-5387776kn^3-1363968kn^5-4392960kn^4-9505536k^2n^2-
$$
$$
9778176k^2n^3-3846144k^2n^4-696736k^2-
960464kn+753664k^3n^3+218880k^3
$$
$$
-4165824k^2n-36864k^5+368640k^4n^2+
491520k^4n+181760k^4+935936k^3n+1413120k^3n^2
$$
$$
-16384k^6-49152k^5n-121896k)
\cdot
$$
$$
{{
(-9/4096)
k
} \over {
(4n+3-2k)(4n+1-2k)(2n-k+2)(2n-k+1)(8n+7+2k)(8n+5+2k)(8n+3+2k)}}  
\quad \halmos
$$

Theorem 38 : 
$$
F(-2n \, , \, -2n-1/4 \, , \, 4n+11/4 \, , \, 1/9 ) \, = \,
(65536/59049)^n \, \cdot \,
{{(11/16)_n(15/16)_n(19/16)_n(23/16)_n} \over
{(4/3)_n(5/6)_n(19/24)_n(31/24)_n}} .
$$

Proof : 
$$
(6391980+44661417n+128402584n^2+194585468n^3+163941024n^4+72772864n^5+
13285376n^6-18793632kn^2
$$
$$
-17944128kn^3-1363968kn^5-8184832kn^4
-17754048k^2n^2-13460480k^2n^3
$$
$$
-3846144k^2n^4-2275184k^2-
9616576kn+753664k^3n^3+846464k^3
-10406208k^2n-69632k^5
$$
$$
+368640k^4n^2+644096k^4n+281088k^4+2419968k^3n+
2332672k^3n^2-16384k^6-49152k^5n-1949232k)
\cdot
$$
$$
{{
(-9/128)
k
} \over {
(8n+9-4k)(8n+5-4k)(2n-k+2)(2n-k+1)(16n+19+4k)(16n+15+4k)(16n+11+4k)}}  
\quad \halmos
$$

Theorem 39 : Let $\zeta \, = \, -1 \pm \sqrt{2}$, then
$$
F(-2n \, , \, 1/2 \, , \, -4n \, , \, 2\zeta  ) \, = \,
{{(3/8)_n(5/8)_n} \over
{(1/4)_n (3/4)_n }} \quad.
$$

Proof : 
$$
{{
(-1/2)
(3+\zeta )
(6\zeta n-2n+4\zeta -3k^2-k^2\zeta +8kn+6k)(4n-k+1)k
} \over {
(2n-k+2)(2n-k+1)(8n+5)(8n+3)}}  
\quad \halmos
$$

Theorem 40 : 
$$
F(-2n \, , \, 1/2 \, , \, 4n+5/2 \, , \, 1/4 ) \, = \,
{{(7/8)_n(5/8)_n(9/8)_n(11/8)_n} \over
{(3/4)_n^2 (5/4)_n^2}} \quad .
$$

Proof : 
$$
{{
(-4/3)
(1+2k)(256n^3+640n^2-24k^2n+550n-4k^3-24k^2+k+162)k
} \over {
(2n-k+2)(2n-k+1)(8n+9+2k)(8n+7+2k)(8n+5+2k)}}  
\quad \halmos
$$

Theorem 41 : 
$$
F(-2n \, , \, 1/2 \, , \, 4n+3/2 \, , \, 1/4 ) \, = \,
{{(3/8)_n(7/8)_n(5/8)_n(9/8)_n} \over
{(7/12)_n (5/12)_n(11/12)_n(13/12)_n}} \quad.
$$

Proof : 
$$
{{
(-4/81)(-1+6k)(2304n^3+4608n^2-216k^2n+144kn+3214n-36k^3-144k^2+117k+770)k
} \over
{
(2n-k+2)(2n-k+1)(8n+7+2k)(8n+5+2k)(8n+3+2k)}}  
\quad \halmos
$$

Theorem 42 : 
Let $\omega$ be a primitive cubic root of unity
(i.e. $\omega \, = \, (1 \pm \sqrt{3} i)/2$), then
$$
F(-2n \, , \, 2n+1 \, ,\,  -4n \, , \, \omega  ) \, = \,
((27/16) \omega )^n \, \cdot \,
{{(1/3)_n(5/6)_n} \over
{(1/4)_n(3/4)_n}} \quad .
$$
Proof : 
$$
(2660+14784n+30156n^2+26880n^3+8848n^4+308\omega +1316\omega n-
11k^4\omega -16k^4-1936kn^3\omega -4852kn^2\omega -4086kn\omega 
$$
$$
+204k^3+1234k^2\omega n+708k^2\omega n^2-526k^2-6052kn^2-2816kn^3+
236k^3n-4234kn+2k^3\omega -1148\omega k+537\omega k^2-780k^2n^2
$$
$$
+24k^3\omega n-1322k^2n+1792\omega n^2+784\omega n^3-966k)
\cdot
$$
$$
{{
(1/6636)k(4n-k+1)
(-27+11\omega ) } \over {
(2n+1)(2n-k+2)(2n-k+1)(6n+5)(3n+1)(n+1)}}  
\quad \halmos
$$

Theorem 43 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$ then
$$
F(-2n \, , \, 2n+1 \, , \, -3n+1/2 \, , \, \phi  ) \, = \,
((100/27)\phi +25/9)^n \, \cdot \,
{{(3/10)_n(7/10)_n} \over
{(1/6)_n(5/6)_n}} \quad .
$$

Proof : 
$$
(1678n+2708n^2+1448n^3+136k^2\phi n-264k\phi n^2-364k\phi n-
118k\phi +324\phi n
$$
$$
+264\phi n^2-26k^3\phi +96k^2\phi +343+96\phi -
267k-19k^2+24k^3-42k^2n-536kn^2-750kn)
\cdot
$$
$$
{{
(2/905)k(6n+1-2k)(-43+16\phi )
} \over {
(2n+1)(2n-k+2)(2n-k+1)(10n+7)(10n+3)}}  \quad \halmos
$$

Theorem 44 : Let $\phi$ be the Golden ratio $(1+\sqrt{5})/2$ then
$$
F(-2n \, , \, 2n+1 \, , \, -3n \, , \, \phi  ) \, = \,
(100/27\phi +25/9)^n \, \cdot \,
{{(2/5)_n(3/5)_n} \over {(1/3)_n(2/3)_n}} .
$$

Proof : 
$$
(1334n+1716n^2+724n^3-270k\phi n+68k^2\phi n-132k\phi n^2-125k\phi +
214\phi n+132\phi n^2
$$
$$
-13k^3\phi +68k^2\phi +82\phi -177k-21k^2+12k^3-
21k^2n-268kn^2-433kn+342)
\cdot
$$
$$
{{
(1/905)k(3n-k+1)
(-43+16\phi )
} \over {
(2n-k+2)(2n-k+1)(5n+3)(5n+2)(n+1)}}  \quad \halmos
$$

Theorem 45 : Let $\zeta_1 \, = \, 1 \pm \sqrt{2}$, then
$$
F(-2n \, , \, 2n+1 \, , \, -2n+1/2 \, , \, (1/2)\zeta_1  ) \, = \,
(4+8\zeta_1 )^n \, \cdot \,
{{(3/8)_n(5/8)_n} \over {(1/4)_n(3/4)_n}} \quad .
$$

Proof : 
$$
(56n^2+72n+8\zeta_1 n+24+5\zeta_1 -6k^2-16kn-12k+4k^2\zeta_1 -
8\zeta_1 kn-6\zeta_1 k)
\cdot
$$
$$
{{
(2/7)k(4n+1-2k)
(-4+\zeta_1 )
} \over {
(2n-k+2)(2n-k+1)(8n+5)(8n+3)}}  \halmos
$$

Theorem 46 : Let $\alpha \, = \, 2 \pm \sqrt{3}$, then
$$
F(-2n \, , \, 2n+1 \, , \, -n+3/4 \, , \,  (1/4)
\alpha  )=(3-6\alpha )^n \, \cdot \,
{{(5/12)_n} \over {(1/4)_n}} \quad .
$$

Proof : 
$$
{{
(1/3)(8n+5+\alpha -2k\alpha +2k)
(-5+\alpha )(4n+1-4k)k } \over 
{(2n-k+2)(2n-k+1)(12n+5)}}  \quad \halmos
$$

Theorem 47 : 
$$
F(-2n \, , \, 3n+1 \, , \, -2n+2 \, , \, 2/5 ) \, = \,
(125/27)^n \, \cdot \, {{(1/2)_n} \over {(-1/2)_n}} \quad.
$$

Proof : 
$$
{{
(-1/25)(147n^2+33nk+179n+3k^2+23k+50)(2n-k-1)k} \over
{n(2n+2-k)(3n+2)(3n+1)}}  \quad \halmos
$$

Theorem 48 : 
$$
F(-2n \, , \, 4n+1 \, , \, -4n+1/2 \, , \, 2 ) \, = \,
16^n \, \cdot \,
{{(5/16)_n(3/16)_n(13/16)_n(11/16)_n} \over
{(3/8)_n(1/8)_n(7/8)_n(5/8)_n}} \quad .
$$

Proof : 
$$
(20886+229992n+1021408n^2+2335744n^3+2908160n^4
$$
$$
+1875968n^5+491520n^6-
852992kn^4-1028864kn^3-278528kn^5
$$
$$
-611776kn^2+86016k^2n^3+30720k^2n^4+
43152k^2n-5344k^3n-5760k^3n^2
$$
$$
-2048k^3n^3-1583k^3+91168k^2n^2
$$
$$
+278k^4+480k^4n+128k^4n^2-68k^5+8k^6-64k^5n+7628k^2-179848nk-21029k)
\cdot
$$
$$
{{
(-1/4)
(8n+1-2k)k
} \over {
(2n-k+2)(2n-k+1)(16n+11)(16n+13)(16n+3)(16n+5)(4n+3)(4n+1)}}  
\quad \halmos
$$

Theorem 49 : 
$$
F(-2n \, , \, 4n+2 \, , \, -4n-1/2 \, , \, 2 ) \, = \,
16^n \, \cdot \,
{{(9/16)_n(7/16)_n(15/16)_n(17/16)_n} \over
{(3/8)_n(7/8)_n(5/8)_n(9/8)_n}} \quad.
$$

Proof : 
$$
(268890+1832424n+5143712n^2+7618816n^3+6283264n^4+2736128n^5+
491520n^6-1309696kn^4
$$
$$
-2438912kn^3-278528kn^5-2249600kn^2+
125952k^2n^3+30720k^2n^4+134312k^2n-10400k^3n-8064k^3n^2-2048k^3n^3-
$$
$$
4313k^3+194848k^2n^2+518k^4+640k^4n+128k^4n^2-92k^5+8k^6-64k^5n+34640k^2-
1028112nk-186251k) \cdot
$$
$$
{{
(-1/4)
(8n+3-2k)k} \over {
(2n-k+2)(2n-k+1)(16n+17)(16n+15)(16n+7)(16n+9)(4n+5)(4n+3)}}  
\quad \halmos
$$

Theorem 50 : 
$$
F(-2n \, , \, 4n+1 \, , \, -2n+2 \, , \, 1/3 ) \, = \,
(81/16)^n \, \cdot \,
{{(1/2)_n} \over {(-1/2)_n}} .
$$

Proof : 
$$
(1040n^3+1812n^2+264kn^2+982n+36k^2n+342kn+24k^2+162+103k+2k^3) \cdot
$$
$$
{{
(-1/54)
(2n-k-1)k}
\over
{n(2n-k+2)(4n+3)(2n+1)(4n+1)}}  \quad \halmos
$$

Theorem 51 : Let $\zeta_2 \, = \, 1 \pm \sqrt{3}/3$, then

$$
F(-2n \, , \, 4n+2 \, , \, 4/3 \, , \, (2/3)\zeta_2  ) \, = \, 
(-9+18\zeta_2 )^n \, \cdot \,
{{(5/12)_n} \over {(5/4)_n }} \quad .
$$

Proof : 
$$
(352n^2+432n+168\zeta_2 n+116+138\zeta_2 -2k^2+9\zeta_2 k^2-
$$
$$
288\zeta_2 nk+152nk+126k-237\zeta_2 k)
\cdot
$$
$$
{{
(-1/88)
(-16+9\zeta_2 )
(1+3k)k } \over {
(2n-k+2)(2n-k+1)(12n+5)(4n+3)}}  \quad \halmos
$$

Theorem 52 : Let $\zeta_2 \, = \, 1 \pm \sqrt{3}/3$, then
$$
F(-2n \, , \, 4n+1 \, , \, 2/3 \, , \, (2/3)\zeta_2  ) \, = \,
(-9+18\zeta_2 )^n \, \cdot \,
{{(7/12)_n} \over {(3/4)_n}} \quad .
$$

Proof : 
$$
(352n^2+264n+264\zeta_2 n+20+174\zeta_2 -2k^2+9\zeta_2 k^2-
288\zeta_2 nk+152nk+102k-195\zeta_2 k)
\cdot
$$
$$
{{
(-1/88)
(-16+9\zeta_2 )
(-1+3k)k
} \over {
(2n-k+2)(2n-k+1)(12n+7)(4n+1)}}  \quad \halmos
$$





\end