By a process not obvious how to automate, I extracted HypergeometricPFQ[{a, b, (2 a)/5 + (6 b)/5 + 1, 2 a + 2 b, a + 3 b}, {(2 a)/5 + (6 b)/5, a + (3 b)/2 + 1/2, a + (3 b)/2 + 1, 2 b + 1}, -(1/4)] == (4^-a Gamma[b + 1/2] Gamma[2 a + 3 b + 1])/( Gamma[a + b + 1/2] Gamma[a + 3 b + 1]) from a summation containing a messy, irreducible cubic factor, analogous to the sort used to discover recurrence relations via "creative telescopy". Punting the cubic cost two degrees of freedom, but there were hundreds of variants. The unpleasant choice seems to be between a useless messy cubic, and a nearly useless identity with only two degrees of freedom. Probably the identity you need was among the hundreds I threw away. Here's an interesting remedy. What you actually tabulate is, in this case, the fully general result in a somewhat intimidating form: ((-b - d) ((1/(-1 + b))(-1 + b - d) (-1 + a + b - d) (-1 + b + c - d) HypergeometricPFQ[{a, -1 + b, c, d, -1 - a + b + d, -1 + b - c + d}, {b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)] - (19 + 5 b^2 - 7 c + b (-19 + 4 c - 11 d) + a (-7 + 4 b + c - 3 d) + 20 d - 3 c d + 5 d^2) HypergeometricPFQ[{a, b, c, d, -1 - a + b + d, -1 + b - c + d}, {b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)] - 5 (-3 + b) (-2 + b) HypergeometricPFQ[{a, -1 + b, b, c, d, -1 - a + b + d, -1 + b - c + d}, {-3 + b, b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)] + 3 (-2 + b) (-8 + a + 3 b + c - 3 d) HypergeometricPFQ[{a, -1 + b, b, c, d, -1 - a + b + d, -1 + b - c + d}, {-2 + b, b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)]))/(a c (-1 + b + c) (1 - b + c - d) d) == -((Gamma[1 - a + d] Gamma[1 + b + d] Gamma[ 1 - c + d] Gamma[-a + b - c + d])/(a (-1 + b) c (-1 + b + c) Gamma[1 + d] Gamma[-1 - a + b + d] Gamma[ 1 - a - c + d] Gamma[b - c + d])) It's then fairly routine to determine whether your pFq[-1/4] is a special case. It's not pencil and paper, but many people now have access to computers. --rwg Count yourself lucky if you can read "Rauzy fractal" and not remember Them Moose Goosers.