52 Strange Evaluations by Shalosh B. Ekhad

Theorem 1 :

F (−n , −4n − 1/2 , −3n , −1) = (64/27)^{n} · \frac{(3/8)_{n}(5/8)_{n}}{(1/3)_{n}(2/3)_{n}} .

Proof :

\frac{−2(296n^{3} − 216n^{2}k + 771n^{2} + 651n − 372nk + 52nk^{2} − 155k + 177 − 4k^{3} + 45k^{2})(3n − k + 1)k}{(8n + 9 − 2k)(8n + 7 − 2k)(8n + 5 − 2k)(8n + 3 − 2k)(n − k + 1)}

Theorem 2 :

F (−n , −4n − 5/2 , −3n − 1 , −1) = (64/27)^{n} · \frac{(7/8)_{n}(9/8)_{n}}{(2/3)_{n}(4/3)_{n}} .

Proof :

(296n^{3} + 1215n^{2} − 216n^{2}k + 1660n + 52nk^{2} − 586nk + 756 − 394k − 4k^{3} + 71k^{2})·

\frac{−2(3n − k + 2)k}{(8n + 13 − 2k)(8n + 11 − 2k)(8n + 9 − 2k)(8n + 7 − 2k)(n − k + 1)}

Theorem 3 :

F (−n , −3n − 1/2 , −4n , 4) = (−27/16)^{n} · \frac{(1/3)_{n}(5/6)_{n}}{(1/4)_{n}/(3/4)_{n}} .

Proof :

\frac{(344n^{3} − 216n^{2}k + 768n^{2} + 522n − 282nk + 34nk^{2} − 75k + 104 + 13k^{2})(4n − k + 1)k}{4(6n + 7 − 2k)(6n + 5 − 2k)(6n + 3 − 2k)(n − k + 1)(3n + 1)} .

Theorem 4 :

F (−n , −3n − 3/4 , −4n − 1 , −8) = (27/4)^{n} · \frac{(2/3)_{n}(7/12)_{n}}{(1/2)_{n}(3/4)_{n}} .

Proof :

\frac{−(2208n^{3} − 1488n^{2}k + 6272n^{2} + 5762n − 2620nk + 252nk^{2} − 1082k + 1707 + 167k^{2})(4n − k + 2)k}{12(12n + 15 − 4k)(12n + 11 − 4k)(12n + 7 − 4k)(n − k + 1/(3n + 2)}

Theorem 5 :

F (−n , −3n − 3/2 , −4n − 2 , −8) = (27/4)^{n} · \frac{(5/6)_{n}(7/6)_{n}}{(3/4)_{n}(5/4)_{n}} .

Proof :

−(1/24)\frac{(184n^{2} + 492n − 124nk + 326 + 21k^{2} − 167k)(4n − k + 3)k}{(6n + 9 − 2k)(6n + 7 − 2k)(6n + 5 − 2k)(n − k + 1)}

Theorem 6 :

F (−n , −3n − 1/4 , −4n , −8) = (27/4)^{n} · \frac{(1/3)_{n}(5/12)_{n}}{(1/2)_{n}/(1/4)_{n}} .

Proof :

\frac{(−1/4)(736n^{3} − 496n^{2}k + 1600n^{2} + 1058n + 84nk^{2} − 624nk − 149k + 200 + 27k^{2})(4n − k + 1)k}{(12n + 13 − 4k)(12n + 9 − 4k)(12n + 5 − 4k)(n − k + 1)(3n + 1)}

Theorem 7 :

F (−n , −3n − 1/2 , −4n , −8) = (27/4)^{n} · \frac{(1/3)_{n}(2/3)_{n}}{(1/4)_{n}(3/4)_{n}} .

Proof : (−1/32)(2208n^{4}−1488n^{3}k+6640n^{3}+252n^{2}k^{2}−2988n^{2}k+7104n^{2}+248nk^{2}−1812nk+3156n+53k^{2}−321k+478)·

\frac{(4n − k + 1)k}{(6n + 7 − 2k)(6n + 5 − 2k)(6n + 3 − 2k)(n − k + 1)(3n + 2)(3n + 1)}

Theorem 8 :

F (−n , −3n − 5/4 , −4n − 1 , −8) = (27/4)^{n} · \frac{(2/3)_{n}(13/12)_{n}}{(1/2)_{n}(5/4)_{n}} .

Proof :

\frac{(−1/4)(736n^{3} − 496n^{2}k + 2336n^{2} + 2382n − 956nk + 84nk^{2} − 414k + 767 + 55k^{2})(4n − k + 2)k}{(12n + 17 − 4k)(12n + 13 − 4k)(12n + 9 − 4k)(n − k + 1)(3n + 2)}

Theorem 9 :

F (−n , −3n − 3/2 , −4n − 2 , 4) = (−27/16)^{n} · \frac{(4/3)_{n}(5/6)_{n}}{(3/4)_{n}(5/4)_{n}} .

Proof :

\frac{(1/12)(1032n^{3} − 648n^{2}k + 4024n^{2} + 5202n − 1690nk + 102nk^{2} − 1103k + 2228 + 135k^{2})(4n − k + 3)k}{(6n + 9 − 2k)(6n + 7 − 2k)(6n + 5 − 2k)(n − k + 1)(3n + 4)}

Theorem 10 :

F (−n , −3n − 7/4 , −4n − 2 , −8) = (27/4)^{n} · \frac{(4/3)_{n}(11/12)_{n}}{(3/2)_{n}(3/4)_{n}} .

Proof :

\frac{(−1/12)(2208n^{3} + 9216n^{2} − 1488n^{2}k + 12806n + 252nk^{2} − 4112nk + 5924 − 2839k + 335k^{2})(4n − k + 3)k}{(12n + 19 − 4k)(12n + 15 − 4k)(12n + 11 − 4k)(n − k + 1)(3n + 4)}

Theorem 11 : Let ω be a primitive cubic root of unity (i.e. ω = (1 ± \sqrt{3}i)/2), then

F (−n , −3n − 1 , −2n , ω) = (−(3/2)ω + 3/4)^{n} · \frac{(2/3)_{n})}{(1/2)_{n}} .

Proof :

(129n^{2} + 289n − 2ωn + 160 − 2ω − 88kn + 14kωn − 101k + 19kω + 13k^{2} − 5ωk^{2})·

\frac{(−1/129)(−5 + 13ω)(2n − k + 1)k}{(3n + 4 − k)(3n + 3 − k)(3n + 2 − k)(n − k + 1)}

Theorem 12 :

F (−n , −3n − 1/2 , −n + 1/2 , 4) = (−27)^{n}.

Proof :

\frac{(1/9)k(2n + 1 − 2k)(84n^{2} + 172n − 60nk + 12k^{2} − 64k + 87)}{(6n + 7 − 2k)(6n + 5 − 2k)(6n + 3 − 2k)(n + 1 − k)}

Theorem 13 :

F (−n , −3n − 3/2 , −n + 1/2 , 4) = (−27)^{n} · \frac{(5/6)_{n}(7/6)_{n}}{(1/2)_{n}(3/2)_{n}} .

Proof :

\frac{(1/3)(28n^{2} + 76n − 20nk + 51 − 28k + 4k^{2})(2n + 1 − 2k)k}{(6n + 9 − 2k)(6n + 7 − 2k)(6n + 5 − 2k)(n + 1 − k)}

Theorem 14 : Let α = 9 ± 4\sqrt{5}, then

F (−n , −3n − 1/2 , 2n + 3/2 , α) = (128/125 + 128/125α)^{n} · \frac{(3/4)_{n}(5/4)_{n}}{(4/5)_{n}(6/5)_{n}} .

Proof : (79794 + 388827n + 706989n^{2} + 566136n^{3} + 167796n^{4} − 4518k − 4875nk − 51972k^{2} + 11429αnk+ 3754αk+2132αk^{2}+496k^{4}α+26472k^{3}−3696k^{4}−6816kn^{3}−6174kn^{2}−127434k^{2}n−72852k^{2}n^{2}+30768k^{3}n −3336n^{3}α−9466αn^{2}−8839αn−2718α+10998kαn^{2}+1060k^{2}αn^{2}−2192k^{3}αn+3336kn^{3}α+3058k^{2}αn−2584k^{3}α)·

\frac{−(1/74576)(31α − 327)k}{(6n + 7 − 2k)(6n + 5 − 2k)(6n + 3 − 2k)(n − k + 1)(4n + 3 + 2k)(α − 1)}.

Theorem 15 : Let α = 9 ± 4\sqrt{5}, then

F (−n , −3n − 3/2 , 2n + 5/2 , α) = (128/125 + 128/125α)^{n} · \frac{(5/4)_{n}(7/4)_{n}}{(7/5)_{n}(8/5)_{n}} .

Proof : (384115+1282296n+1575413n^{2}+845796n^{3}+167796n^{4}−86193k−153105nk−77610k^{2}+26963αnk+

13383αk + 730αk^{2} + 496k^{4}α + 34264k^{3} − 3696k^{4} − 6816kn^{3}− 74602kn^{2} − 154458k^{2}n − 72852k^{2}n^{2} + 30768k^{3}n − 3336n^{3}α − 12126αn^{2} − 14517αn− 5745α + 17114kαn^{2} + 1060k^{2}αn^{2} − 2192k^{3}αn + 3336kn^{3}α + 1842k^{2}αn − 2984k^{3}α)·

\frac{k(−1/74576)(31α − 327)}{(6n + 9 − 2k)(6n + 7 − 2k)(6n + 5 − 2k)(n − k + 1)(4n + 5 + 2k)(α − 1)}

Theorem 16 : Let α = 2 ± \sqrt{3}, then

F (−n , −2n − 1/2 , −4n − 1 , 4α) = (3/4 − 3/2α)^{n} · \frac{(2/3)_{n}}{(1/2)_{n}} .

Proof :

(488n^{2} + 978n − 20αn + 472 − 23α + 94k^{2} − 25k^{2}α − 420kn + 52knα − 422k + 72kα)·

\frac{(1/732)(−68 + 19α)(4n + 2 − k)k}{(4n + 5 − 2k)(4n + 3 − 2k)(n − k + 1)(3n + 2)}

Theorem 17 :

F (−n , −2n − 1/3 , −3n , 9) = (−4)^{n} · \frac{(1/2)_{n}}{(1/3)_{n}} .

Proof :

\frac{(1/9)k(3n − k + 1)(45n^{2} − 21nk + 72n − 11k + 25)}{(6n + 7 − 3k)(6n + 4 − 3k)(n − k + 1)(2n + 1)} .

Theorem 18 :

F (−n , −2n − 2/3 , −3n − 1 , 9) = (−4)^{n} · \frac{(5/6)_{n}}{(2/3)_{n}} .

Proof :

\frac{(1/6)(15n + 19 − 7k)(3n − k + 2)k}{(6n + 8 − 3k)(6n + 5 − 3k)(n − k + 1)}

Theorem 19 :

F (−n , −n − 1/2 , 4n + 9/2 , 1/5) = (\frac{16384}{15625})^{n} · \frac{(9/8)_{n}(11/8)_{n}(13/8)_{n}(15/8)_{n}}{(6/5)_{n}(9/5)_{n}(13/10)_{n}(17/10)_{n}} .

Proof :

(3036n^{4} + 1256n^{3}k + 16960n^{3} + 35275n^{2} + 4780n^{2}k − 1184n^{2}k^{2} + 6180nk− 3440nk^{2} − 544nk^{3} + 32445n + 2720k + 11154 − 2480k^{2} − 64k^{4} − 800k^{3})·

\frac{(−5/16)k}{(2n + 3 − 2k)(n − k + 1)(8n + 13 + 2k)(8n + 11 + 2k)(8n + 9 + 2k)}

Theorem 20 :

F (−n , −n + 1/2 , 4n + 3/2 , 1/5) = (\frac{16384}{15625})^{n} · \frac{(3/8)_{n}(7/8)_{n}(5/8)_{n}(9/8)_{n}}{(3/5)_{n}(2/5)_{n}(9/10)_{n}(11/10)_{n}} .

Proof :

(3036n^{4} + 1256n^{3}k + 7852n^{3} + 7409n^{2} + 2844n^{2}k − 1184n^{2}k^{2} + 2252nk − 1344nk^{2} + 2993n−

544nk^{3} − 336k^{2} + 592k + 430 − 352k^{3} − 64k^{4})·

\frac{(−5/16)k}{(2n + 1 − 2k)(n − k + 1)(8n + 7 + 2k)(8n + 5 + 2k)(8n + 3 + 2k)}

Theorem 21 : Let β = (3 ± sqrt5)/2, then

F (−n , 2n + 1 , −3n , β) = ((25/27) + (25/27)β)^{n} · \frac{(2/5)_{n}(3/5)_{n}}{(1/3)_{n}(2/3)_{n}} .

Proof :

(1376n + 308 + 2026n^{2} + 958n^{3} − 60k − 23k^{3}β + 38k^{2}β − 91kβ + 139βn + 87βn^{2} −201kβn − 87kβn^{2} + 38k^{2}βn + 48k^{3} + 4k^{2} − 122kn − 110kn^{2} + 4k^{2}n + 52β)·

\frac{(1/4790)k(3n − k + 1)(−78 + 17β)}{(n − k + 1)(5n + 3)(5n + 2)(n + 1)(2n + 1)}

Theorem 22 : Let β = (3 ± sqrt5)/2, then

F (−n , 2n + 2 , −3n − 1 , β) = (25/27 + 25/27β)^{n} · \frac{(4/5)_{n}(6/5)_{n}}{(2/3)_{n}(4/3)_{n}} .

Proof : (4179n + 3463n^{2} + 958n^{3} + 24k − 317kβn − 87kβn^{2} + 48k^{3} + 54k^{2} − 109kn − 110kn^{2} + 4k^{2}n + 96β+

1674 + 183βn + 87βn^{2} + 38k^{2}βn − 23k^{3}β + 34k^{2}β − 251kβ)·

\frac{(1/4790)k(3n − k + 2)(−78 + 17β)}{(n − k + 1)(5n + 6)(5n + 4)(2n + 3)(n + 1)}

Theorem 23 : Let ϕ be the Golden ratio (1 + \sqrt{5})/2 then

F (−n , 2n + 1 , 4n + 2 , ϕ^{−1}) = (128/125 − 64/125ϕ^{−1})^{n} · \frac{(3/4)_{n}(5/4)_{n}}{(4/5)_{n}(6/5)_{n}} .

Proof :

(2815n + 3661n^{2} + 1558n^{3} + 14k^{3}ϕ^{−1} − 148ϕ^{−1}n − 84ϕ^{−1}n^{2} + 579k + 183kϕ^{−1}n+
1567kn + 1013kn^{2} + 88kϕ^{−1} + 37k^{2}ϕ^{−1}n + 34k^{2}ϕ^{−1}+
270k^{2}n + 84kϕ^{−1}n^{2} + 712 + 206k^{2} + 39k^{3} − 64ϕ^{−1})·

\frac{(−1/779)k(14ϕ^{−1} − 25)}{(n − k + 1)(4n + 4 + k)(4n + 3 + k)(4n + 2 + k)(−1 + ϕ^{−1})}

Theorem 24 : Let ϕ be the Golden ratio (1 + \sqrt{5})/2 then

F (−n , 2n + 2 , 4n + 4 , ϕ^{−1}) = (128/125 − 64/125ϕ^{−1})^{n} · \frac{(5/4)_{n}(7/4)_{n}}{(7/5)_{n}(8/5)_{n}} .

Proof :

(7628n + 5998n^{2} + 1558n^{3} + 14k^{3}ϕ^{−1} − 178ϕ^{−1}n
−84ϕ^{−1}n^{2} + 3188 + 1762k + 309kϕ^{−1}n + 2697kn + 1013kn^{2} + 253kϕ^{−1}+

37k^{2}ϕ^{−1}n + 79k^{2}ϕ^{−1} + 270k^{2}n + 84kϕ^{−1}n^{2} + 387k^{2} + 39k^{3} − 94ϕ^{−1})·

\frac{(−1/779)k(14ϕ^{−1} − 25)}{(n − k + 1)(4n + 6 + k)(4n + 5 + k)(4n + 4 + k)(−1 + ϕ^{−1})}

Theorem 25 : Let γ = 3 ± 2\sqrt{2}, then

F (−2n , −4n − 1/2 , −2n + 1/2 , γ) = (−16 + 96γ)^{n} \frac{(3/8)_{n}(5/8)_{n}}{(1/4)_{n}(3/4)_{n}} .

Proof :

(25434 + 137162n + 271600n^{2} + 233440n^{3} + 73472n^{4} + 5216k^{2}γn^{2} + 10144k^{2}γn − 21148kγn−

6208kγn^{3} − 20304kγn^{2} − 1264k^{3}γn − 142224kn^{2} − 58944kn^{3} − 111324kn + 14304k^{2}n^{2}+ 22536k^{2}n − 1776k^{3}n − 1372k^{3} + 8594k^{2} + 88k^{4} + 4834k^{2}γ − 7067kγ − 1204k^{3}γ − 28145k + 3837γ+

104k^{4}γ + 13852γn + 6208γn^{3} + 16240γn^{2})·

\frac{(−1/1148)(−89 + 13γ)(4n + 1 − 2k)k}{(8n + 9 − 2k)(8n + 7 − 2k)(8n + 5 − 2k)(8n + 3 − 2k)(2n − k + 1)(2n − k + 2)(γ − 1)}

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^{2}n^{2} + 290142k^{2}n + 119151k^{2} − 32076k^{3}n − 26730k^{3} + 2187k^{4})·

\frac{(1/729)(−1 + 3k)k}{(2n − k + 1)(2n − k + 2)(12n + 4 − 3k)(12n + 7 − 3k)(12n + 10 − 3k)(12n + 13 − 3k)}

Theorem 27 : Let δ = 5 ± 3\sqrt{3}, then

F (−2n , −4n − 1/3 , 2/3 , 2δ) = (702δ + 135)^{n} · \frac{(7/12)_{n}}{(3/4)_{n}} .

Proof :

(2059512 + 12099448n + 766292δn + 25671232n^{2} + 23410464n^{3} + 7767936n^{4} + 933000δn^{2}− 1239770kδn − 404352kn^{3}δ + 506562k^{2}δn + 268020k^{2}δn^{2} − 62100k^{3}δn − 1257180kδn^{2} −2121648k − 5589216kn^{3} − 12542136kn^{2} − 9090980kn + 1684440k^{2}n^{2} + 2508684k^{2}n

−240624k^{3}n − 56934k^{3}δ + 232953k^{2}δ − 390660kδ + 370656n^{3}δ
+205020δ + 13446k^{4} − 178812k^{3} + 904542k^{2} + 4941k^{4}δ)·

\frac{(1/364122)k(−1 + 3k)(61δ − 776)}{(12n + 13 − 3k)(12n + 10 − 3k)(12n + 7 − 3k)(12n + 4 − 3k)(2n − k + 1)(2n − k + 2)2δ − 1)}

Theorem 28 : Let δ = 5 ± 3\sqrt{3}, then

F (−2n , −4n − 2/3 , 4/3 , 2δ) = (702δ + 135)^{n} · \frac{(5/12)_{n}}{(5/4)_{n}} .

Proof :

(3710744 + 18504312n + 607916δn + 33746032n^{2} + 26726016n^{3} + 7767936n^{4} + 661656δn^{2}− 1315958kδn − 404352kn^{3}δ + 533802k^{2}δn + 268020k^{2}δn^{2} − 62100k^{3}δn − 1280676kδn^{2}− 3169096k − 5589216kn^{3} − 14213784kn^{2} − 11770892kn + 1684440k^{2}n^{2} + 2829540k^{2}n − 240624k^{3}n

−60696k^{3}δ + 260835k^{2}δ − 439228kδ + 235872n^{3}δ + 182668δ

+13446k^{4} − 200988k^{3} + 1159494k^{2} + 4941k^{4}δ)·

\frac{(1/364122)k(1 + 3k)(61δ − 776)}{(12n + 14 − 3k)(12n + 11 − 3k)(12n + 8 − 3k)(12n + 5 − 3k)(2n − k + 1)(2n − k + 2)(2δ − 1)}

Theorem 29 :

F (−2n , −4n − 2/3 , 4/3 , −8) = 729^{n} · \frac{(5/12)_{n}(11/12)_{n}}{(3/4)_{n}(5/4)_{n}} .

Proof :

(34944n^{4} − 43776kn^{3} + 131104n^{3} − 121596kn^{2} + 181768n^{2} + 19116k^{2}n^{2}−

110586kn + 110164n + 35046k^{2}n − 3564k^{3}n + 15741k^{2} + 243k^{4} − 32868k + 24564 − 3240k^{3})·

\frac{(1/81)(1 + 3k)k}{(12n + 14 − 3k)(12n + 11 − 3k)(12n + 8 − 3k)(12n + 5 − 3k)(2n − k + 1)(2n − k + 2)}

Theorem 30 : Let α = 2 ± \sqrt{3}, then

F (−2n , −3n − 1/4 , −4n , 4α) = (39/2α − 21/4)^{n} · \frac{(5/12)_{n}}{(1/4)_{n}} .

Proof :

(171978n + 238000n^{2} + 104736n^{3} − 12496kαn^{2} − 32440kαn + 7780k^{2}αn + 17488αn^{2}+

30900αn + 8407k^{2}α − 1216k^{3}α − 18581kα + 39182 + 13262α

−81808kn^{2} + 15490k^{2} − 1792k^{3} − 43403k − 122224kn + 21112k^{2}n)·

\frac{(−1/3273)k(4n − k + 1)(397α − 1484)}{(12n + 13 − 4k)(12n + 9 − 4k)(12n + 5 − 4k)(2n − k + 1)(2n − k + 2)}

Theorem 31 :

F (−2n , −3n + 1 , −2n + 2 , −2) = 27^{n} · \frac{(1/2)_{n}}{(−1/2)_{n}} .

Proof :

\frac{(−1/9)k(2n − k − 1)(39n^{2} + 55n − 21kn − 14k + 3k^{2} + 18)}{(3n + 2 − k)(3n + 1 − k)(3n − k)(2n + 2 − k)}

Theorem 32 :

F (−2n , −3n , 2n + 1/2 , 1/4) = (27/64)^{n} · \frac{(1/3)_{n}(2/3)_{n}}{(1/4)_{n}(3/4)_{n}} .

Proof :

\frac{(−2/3)(74n^{3} + 168n^{2} − 57kn^{2} − 87kn + 124n + 12k^{2}n − 32k + 30 − k^{3} + 9k^{2})(4n + 1 − 2k)k}{(3n + 3 − k)(3n + 2 − k)(3n + 1 − k)(2n + 1 − k)(2n + 2 − k)}

Theorem 33 : Let η = −7 ± 4\sqrt{3}, then

F (−2n , −3n − 1/4 , −n + 3/4 , η) = (312η + 24)^{n} · \frac{(5/12)_{n}}{(1/4)_{n}} .

Proof :

(485694n + 611008n^{2} + 248736n^{3} − 5600k^{2}ηn + 11504kηn^{2} + 23720ηnk − 11880ηn− 6512ηn^{2} + 752k^{3}η + 22744k^{2} + 11851ηk − 5432ηk^{2} − 204544nk − 79751k − 2416k^{3}

+28576k^{2}n − 126928kn^{2} − 5299η + 124358)·

\frac{(1/124368)(47η + 809)(4n + 1 − 4k)k}{(12n + 13 − 4k)(12n + 9 − 4k)(12n + 5 − 4k)(2n − k + 1)(2n − k + 2)(η − 1)}

Theorem 34 :

F (−2n , −2n + 1/3 , −n + 5/6 , −1/8 ) = (27/16)^{n} · \frac{(1/3)_{n}}{(1/6)_{n}} .

Proof :

\frac{(−4/27)(132n^{2} − 60nk + 202n + 74 + 9k^{2} − 45k)(6n + 1 − 6k)k}{(6n + 5 − 3k)(6n + 2 − 3k)(2n − k + 1)(2n − k + 2)}

Theorem 35 :

F (−2n , −2n + 1/2 , 4n + 3/2 , −1/3) = (4096/6561)^{n} · \frac{(3/8)_{n}(7/8)_{n}(5/8)_{n}(9/8)_{n}}{(1/3)_{n}/(5/6)_{n}/(7/12)_{n}/(13/12)_{n}} .

Proof :

(21483 + 241989n + 1106816n^{2} + 2625708n^{3} + 3409136n^{4} + 2299968n^{5} + 631040n^{6}−

405096kn^{2}−723616kn^{3}−213504kn^{5}−628608kn^{4}−161808k^{2}n^{2}−175872k^{2}n^{3}−72960k^{2}n^{4}−10936k^{2}

−109724kn + 31744k^{3}n^{3} + 5856k^{3} − 67368k^{2}n − 768k^{5} + 5376k^{4}n^{2} + 6720k^{4}n + 2624k^{4}

+31136k^{3}n + 54144k^{3}n^{2} − 256k^{6} − 1536k^{5}n − 11388k)·

\frac{(−3/32)k}{(4n + 3 − 2k)(4n + 1 − 2k)(2n − k + 2)(2n − k + 1)(8n + 7 + 2k)(8n + 5 + 2k)(8n + 3 + 2k)}

Theorem 36 :

F (−2n , −2n + 1/4 , 4n + 5/4 , 1/9) = (65536/59049)^{n} · \frac{(9/16)_{n}(5/16)_{n}(13/16)_{n}(17/16)_{n}}{(1/3)_{n}(5/6)_{n}(13/24)_{n}(25/24)_{n}} .

Proof : (501813+5630279n+25382756n^{2}+59064356n^{3}+75057056n^{4}+49523456n^{5}+13285376n^{6}+1953360kn^{2}

−59200kn^{3} − 1363968kn^{5} − 2306048kn^{4} − 9668544k^{2}n^{2} − 9942016k^{2}n^{3} − 3846144k^{2}n^{4}

−684064k^{2} + 1280440kn + 753664k^{3}n^{3} + 36608k^{3} − 4189920k^{2}n

−16384k^{5}+368640k^{4}n^{2}+523264k^{4}n+195584k^{4}+446720k^{3}n+1058816k^{3}n^{2}−16384k^{6}−49152k^{5}n+247832k)·

\frac{(−9/128)k}{(8n + 7 − 4k)(8n + 3 − 4k)(2n − k + 2)(2n − k + 1)(16n + 13 + 4k)(16n + 9 + 4k)(16n + 5 + 4k)}

Theorem 37 :

F (−2n , −2n + 1/2 , 4n + 3/2 , 1/9) = (65536/59049)^{n} · \frac{(3/8)_{n}(7/8)_{n}(5/8)_{n}(9/8)_{n}}{(7/12)_{n}(5/12)_{n}(11/12)_{n}(13/12)_{n}} .

Proof :

(693315 + 7099938n + 29906648n^{2} + 66047424n^{3} + 80454656n^{4} + 51184128n^{5} + 13285376n^{6}

−3206400kn^{2} − 5387776kn^{3} − 1363968kn^{5} − 4392960kn^{4} − 9505536k^{2}n^{2}−

9778176k^{2}n^{3} − 3846144k^{2}n^{4} − 696736k^{2} − 960464kn + 753664k^{3}n^{3} + 218880k^{3}

−4165824k^{2}n − 36864k^{5} + 368640k^{4}n^{2} + 491520k^{4}n + 181760k^{4} + 935936k^{3}n + 1413120k^{3}n^{2}

−16384k^{6} − 49152k^{5}n − 121896k)·

\frac{(−9/4096)k}{(4n + 3 − 2k)(4n + 1 − 2k)(2n − k + 2)(2n − k + 1)(8n + 7 + 2k)(8n + 5 + 2k)(8n + 3 + 2k)}

Theorem 38 :

F (−2n , −2n − 1/4 , 4n + 11/4 , 1/9) = (65536/59049)^{n} · \frac{(11/16)_{n}(15/16)_{n}(19/16)_{n}(23/16)_{n}}{(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^{2}n^{2} − 13460480k^{2}n^{3} −3846144k^{2}n^{4} − 2275184k^{2} − 9616576kn + 753664k^{3}n^{3} + 846464k^{3} − 10406208k^{2}n − 69632k^{5} +368640k^{4}n^{2}+644096k^{4}n+281088k^{4}+2419968k^{3}n+2332672k^{3}n^{2}−16384k^{6}−49152k^{5}n−1949232k)·

\frac{(−9/128)k}{(8n + 9 − 4k)(8n + 5 − 4k)(2n − k + 2)(2n − k + 1)(16n + 19 + 4k)(16n + 15 + 4k)(16n + 11 + 4k)}

Theorem 39 : Let ζ = −1 ± \sqrt{2}, then

F (−2n , 1/2 , −4n , 2ζ) = \frac{(3/8)_{n}(5/8)_{n}}{(1/4)_{n}(3/4)_{n}} .

Proof :

\frac{(−1/2)(3 + ζ)(6ζn − 2n + 4ζ − 3k^{2} − k^{2}ζ + 8kn + 6k)(4n − k + 1)k}{(2n − k + 2)(2n − k + 1)(8n + 5)(8n + 3)}

Theorem 40 :

F (−2n , 1/2 , 4n + 5/2 , 1/4) = \frac{(7/8)_{n}(5/8)_{n}(9/8)_{n}(11/8)_{n}}{(3/4)_{n}^{2}(5/4)_{n}^{2}} .

Proof :

\frac{(−4/3)(1 + 2k)(256n^{3} + 640n^{2} − 24k^{2}n + 550n − 4k^{3} − 24k^{2} + k + 162)k}{(2n − k + 2)(2n − k + 1)(8n + 9 + 2k)(8n + 7 + 2k)(8n + 5 + 2k)}

Theorem 41 :

F (−2n , 1/2 , 4n + 3/2 , 1/4) = \frac{(3/8)_{n}(7/8)_{n}(5/8)_{n}(9/8)_{n}}{(7/12)_{n}(5/12)_{n}(11/12)_{n}(13/12)_{n}} .

Proof :

\frac{(−4/81)(−1 + 6k)(2304n^{3} + 4608n^{2} − 216k^{2}n + 144kn + 3214n − 36k^{3} − 144k^{2} + 117k + 770)k}{(2n − k + 2)(2n − k + 1)(8n + 7 + 2k)(8n + 5 + 2k)(8n + 3 + 2k)}

Theorem 42 : Let ω be a primitive cubic root of unity (i.e. ω = (1 ± \sqrt{3}i)/2), then

F (−2n , 2n + 1 , −4n , ω) = ((27/16)ω)^{n} · \frac{(1/3)_{n}(5/6)_{n}}{(1/4)_{n}(3/4)_{n}} .

Proof : (2660+14784n+30156n^{2}+26880n^{3}+8848n^{4}+308ω+1316ωn−11k^{4}ω−16k^{4}−1936kn^{3}ω−4852kn^{2}ω−4086knω +204k^{3}+1234k^{2}ωn+708k^{2}ωn^{2}−526k^{2}−6052kn^{2}−2816kn^{3}+236k^{3}n−4234kn+2k^{3}ω−1148ωk+537ωk^{2}−780k^{2}n^{2}

+24k^{3}ωn − 1322k^{2}n + 1792ωn^{2} + 784ωn^{3} − 966k)·

\frac{(1/6636)k(4n − k + 1)(−27 + 11ω)}{(2n + 1)(2n − k + 2)(2n − k + 1)(6n + 5)(3n + 1)(n + 1)}

Theorem 43 : Let ϕ be the Golden ratio (1 + \sqrt{5})/2 then

F (−2n , 2n + 1 , −3n + 1/2 , ϕ) = ((100/27)ϕ + 25/9)^{n} · \frac{(3/10)_{n}(7/10)_{n}}{(1/6)_{n}(5/6)_{n}} .

Proof :

(1678n + 2708n^{2} + 1448n^{3} + 136k^{2}ϕn − 264kϕn^{2} − 364kϕn − 118kϕ + 324ϕn

+264ϕn^{2} − 26k^{3}ϕ + 96k^{2}ϕ + 343 + 96ϕ − 267k − 19k^{2} + 24k^{3} − 42k^{2}n − 536kn^{2} − 750kn)·

\frac{(2/905)k(6n + 1 − 2k)(−43 + 16ϕ)}{(2n + 1)(2n − k + 2)(2n − k + 1)(10n + 7)(10n + 3)}

Theorem 44 : Let ϕ be the Golden ratio (1 + \sqrt{5})/2 then

F (−2n , 2n + 1 , −3n , ϕ) = (100/27ϕ + 25/9)^{n} · \frac{(2/5)_{n}(3/5)_{n}}{(1/3)_{n}(2/3)_{n}} .

Proof :

(1334n + 1716n^{2} + 724n^{3} − 270kϕn + 68k^{2}ϕn − 132kϕn^{2} − 125kϕ + 214ϕn + 132ϕn^{2}

−13k^{3}ϕ + 68k^{2}ϕ + 82ϕ − 177k − 21k^{2} + 12k^{3} − 21k^{2}n − 268kn^{2} − 433kn + 342)·

\frac{(1/905)k(3n − k + 1)(−43 + 16ϕ)}{(2n − k + 2)(2n − k + 1)(5n + 3)(5n + 2)(n + 1)}

Theorem 45 : Let ζ_{1} = 1 ± \sqrt{2}, then

F (−2n , 2n + 1 , −2n + 1/2 , (1/2)ζ_{1}) = (4 + 8ζ_{1})^{n} · \frac{(3/8)_{n}(5/8)_{n}}{(1/4)_{n}(3/4)_{n}} .

Proof :

(56n^{2} + 72n + 8ζ_{1}n + 24 + 5ζ_{1} − 6k^{2} − 16kn − 12k + 4k^{2}ζ_{1} − 8ζ_{1}kn − 6ζ_{1}k)·

\frac{(2/7)k(4n + 1 − 2k)(−4 + ζ_{1})}{(2n − k + 2)(2n − k + 1)(8n + 5)(8n + 3)}

Theorem 46 : Let α = 2 ± \sqrt{3}, then

F (−2n , 2n + 1 , −n + 3/4 , (1/4)α) = (3 − 6α)^{n} · \frac{(5/12)_{n}}{(1/4)_{n}} .

Proof :

\frac{(1/3)(8n + 5 + α − 2kα + 2k)(−5 + α)(4n + 1 − 4k)k}{(2n − k + 2)(2n − k + 1)(12n + 5)}

Theorem 47 :

F (−2n , 3n + 1 , −2n + 2 , 2/5) = (125/27)^{n} · \frac{(1/2)_{n}}{(−1/2)_{n}} .

Proof :

\frac{(−1/25)(147n^{2} + 33nk + 179n + 3k^{2} + 23k + 50)(2n − k − 1)k}{n(2n + 2 − k)(3n + 2)(3n + 1)}

Theorem 48 :

F (−2n , 4n + 1 , −4n + 1/2 , 2) = 16^{n} · \frac{(5/16)_{n}(3/16)_{n}(13/16)_{n}(11/16)_{n}}{(3/8)_{n}(1/8)_{n}(7/8)_{n}(5/8)_{n}} .

Proof :

(20886 + 229992n + 1021408n^{2} + 2335744n^{3} + 2908160n^{4}
+1875968n^{5} + 491520n^{6} − 852992kn^{4} − 1028864kn^{3} − 278528kn^{5}
−611776kn^{2} + 86016k^{2}n^{3} + 30720k^{2}n^{4} + 43152k^{2}n − 5344k^{3}n − 5760k^{3}n^{2}
−2048k^{3}n^{3} − 1583k^{3} + 91168k^{2}n^{2}

+278k^{4} + 480k^{4}n + 128k^{4}n^{2} − 68k^{5} + 8k^{6} − 64k^{5}n + 7628k^{2} − 179848nk − 21029k)·

\frac{(−1/4)(8n + 1 − 2k)k}{(2n − k + 2)(2n − k + 1)(16n + 11)(16n + 13)(16n + 3)(16n + 5)(4n + 3)(4n + 1)}

Theorem 49 :

F (−2n , 4n + 2 , −4n − 1/2 , 2) = 16^{n} · \frac{(9/16)_{n}(7/16)_{n}(15/16)_{n}(17/16)_{n}}{(3/8)_{n}(7/8)_{n}(5/8)_{n}(9/8)_{n}} .

Proof : (268890 + 1832424n + 5143712n^{2} + 7618816n^{3} + 6283264n^{4} + 2736128n^{5} + 491520n^{6} − 1309696kn^{4} −2438912kn^{3}−278528kn^{5}−2249600kn^{2}+125952k^{2}n^{3}+30720k^{2}n^{4}+134312k^{2}n−10400k^{3}n−8064k^{3}n^{2}−2048k^{3}n^{3}− 4313k^{3}+194848k^{2}n^{2}+518k^{4}+640k^{4}n+128k^{4}n^{2}−92k^{5}+8k^{6}−64k^{5}n+34640k^{2}−1028112nk−186251k)·

\frac{(−1/4)(8n + 3 − 2k)k}{(2n − k + 2)(2n − k + 1)(16n + 17)(16n + 15)(16n + 7)(16n + 9)(4n + 5)(4n + 3)}

Theorem 50 :

F (−2n , 4n + 1 , −2n + 2 , 1/3) = (81/16)^{n} · \frac{(1/2)_{n}}{(−1/2)_{n}} .

Proof :

(1040n^{3} + 1812n^{2} + 264kn^{2} + 982n + 36k^{2}n + 342kn + 24k^{2} + 162 + 103k + 2k^{3})·

\frac{(−1/54)(2n − k − 1)k}{n(2n − k + 2)(4n + 3)(2n + 1)(4n + 1)}

Theorem 51 : Let ζ_{2} = 1 ± \sqrt{3}/3, then

F (−2n , 4n + 2 , 4/3 , (2/3)ζ_{2}) = (−9 + 18ζ_{2})^{n} · \frac{(5/12)_{n}}{(5/4)_{n}} .

Proof :

(352n^{2} + 432n + 168ζ_{2}n + 116 + 138ζ_{2} − 2k^{2} + 9ζ_{2}k^{2}−
288ζ_{2}nk + 152nk + 126k − 237ζ_{2}k)·

\frac{(−1/88)(−16 + 9ζ_{2})(1 + 3k)k}{(2n − k + 2)(2n − k + 1)(12n + 5)(4n + 3)}

Theorem 52 : Let ζ_{2} = 1 ± \sqrt{3}/3, then

F (−2n , 4n + 1 , 2/3 , (2/3)ζ_{2}) = (−9 + 18ζ_{2})^{n} · \frac{(7/12)_{n}}{(3/4)_{n}} .

Proof :

(352n^{2} + 264n + 264ζ_{2}n + 20 + 174ζ_{2} − 2k^{2} + 9ζ_{2}k^{2} − 288ζ_{2}nk + 152nk + 102k − 195ζ_{2}k)·

\frac{(−1/88)(−16 + 9ζ_{2})(−1 + 3k)k}{(2n − k + 2)(2n − k + 1)(12n + 7)(4n + 1)}