Conjecture 2F1(14,34;23;13)=1√√4√2−3√4+3√4+4−√2−3√4−2_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}

Using a numerical search on my computer I discovered the following inequality:
|2F1(14,34;23;13)ρ|<1020000,
where ρ is the positive root of the polynomial equation
12ρ812ρ48ρ21=0,
that can be expressed in radicals:
ρ=142341+34+42342.
Based on this inequality I conjecture that the actual difference is the exact zero, i.e.
2F1(14,34;23;13)=ρ.
I looked up in DLMF and MathWorld, but did not find a known special value with exactly these parameters. It also appears that CAS like Maple or Mathematica do not know this identity.


Could you please suggest any ideas how to prove the conjecture (4)?


Update: I can propose even more general conjecture:
27(x1)22F1(14,34;23;x)8+18(x1)2F1(14,34;23;x)482F1(14,34;23;x)2=1

Answer

Let's start with this Pfaff transformation for a=14,b=34,c=23 :
2F1(a,b;c;z)=(1z)a2F1(a,cb;c;zz1)

The 'Darboux evaluation' (42) of Vidunas' "Transformations of algebraic Gauss hypergeometric functions" is :
2F1(14,112;23;x(x+4)34(2x1)3)=(12x)1/4

Solving x(x+4)34(2x1)3=zz1 gives :
z=x(x+4)3(x210x2)2
that we will use as :
z1=4(2x1)3(x210x2)2

while (1) and (2) return :
2F1(14,34;23;z)=[(z1)(2x1)]1/4

so that :

2F1(14,34;23;z)=[4(2x1)4(x210x2)2]1/4
and (up to a minus sign) :
2F1(14,34;23;z)2=x210x22(2x1)2
and indeed the substitution of (z1) and 2F1()2 with (4) and (5) in your formula gives :
27(z1)22F1(14,34;23;z)8+18(z1)2F1(14,34;23;z)482F1(14,34;23;z)2=1

Many other formulae of this kind may be deduced using Vidunas' paper.

Attribution
Source : Link , Question Author : HWᅠ , Answer Author : Raymond Manzoni

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