I've found that this identity seems to be true for all positive integers $n$: $$1= \,_4F_3\left(\frac{1}{6},\frac{5}{6},\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};\frac{3}{2},\frac{5}{6}-\frac{n}{2},\frac{7 }{6}-\frac{n}{2};1\right)2\frac{\left(\frac{n+1}{2}\right)_{n-1} }{\left(\frac{n}{2}+1\right)_{n-1}}\\ -\,_4F_3\left(-\frac{1}{3},\frac{1}{3},\frac{1}{2}-\frac{n}{2},-\frac{n}{2};\frac{1}{2},\frac{1}{3}-\frac{n}{2},\frac{2 }{3}-\frac{n}{2};1\right) $$ I've searched for identities involving the $_4F_3$ hypergeometric function but have not found anything matching this set of parameters.
Note that both hypergeometric functions are Saalschützian and their series are terminating for $n$ a positive integer.
Do you have any ideas of how to prove this is true or find an applicable identity?
The identity shows up in trying to prove that the solution found here: https://mathoverflow.net/questions/291546/second-order-recurrence-relation-for-third-order-polynomial-root is correct using induction. That relation in turn showed up in a quantum field theory calculation.
Best regards, Petter
Mathematica expression for convenience:
0==-1 - HypergeometricPFQ[{-(1/3), 1/3, 1/2 - n/2, -(n/2)}, {1/2,
1/3 - n/2, 2/3 - n/2}, 1] + (
2 HypergeometricPFQ[{1/6, 5/6, 1/2 - n/2, 1 - n/2}, {3/2, 5/6 - n/2,
7/6 - n/2}, 1] Pochhammer[(1 + n)/2, -1 + n])/
Pochhammer[1 + n/2, -1 + n]