Using a numerical search on my computer I discovered the following inequality:

|2F1(14,34;23;13)−ρ|<10−20000,

where ρ is the positive root of the polynomial equation

12ρ8−12ρ4−8ρ2−1=0,

that can be expressed in radicals:

ρ=1√√4√2−3√41+3√4+4−√2−3√4−2.

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 likeMapleorMathematicado 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(x−1)2⋅2F1(14,34;23;x)8+18(x−1)⋅2F1(14,34;23;x)4−8⋅2F1(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)=(1−z)−a2F1(a,c−b;c;zz−1)

The 'Darboux evaluation' (42) of Vidunas' "Transformations of algebraic Gauss hypergeometric functions" is :

2F1(14,−112;23;x(x+4)34(2x−1)3)=(1−2x)−1/4

Solving x(x+4)34(2x−1)3=zz−1 gives :

z=x(x+4)3(x2−10x−2)2

that we will use as :

z−1=4(2x−1)3(x2−10x−2)2

while (1) and (2) return :

2F1(14,34;23;z)=[(z−1)(2x−1)]−1/4

so that :

2F1(14,34;23;z)=[4(2x−1)4(x2−10x−2)2]−1/4

and (up to a minus sign) :

2F1(14,34;23;z)2=−x2−10x−22(2x−1)2

and indeed the substitution of (z−1) and 2F1()2 with (4) and (5) in your formula gives :

27(z−1)2⋅2F1(14,34;23;z)8+18(z−1)⋅2F1(14,34;23;z)4−8⋅2F1(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*