I don't understand why. By subtracting, dividing, and plotting, I sometimes guess the empirical corrections. They're slightly weird: (-2 h + k) Beta[h, k] HypergeometricPFQ[{-h, 1 - (2 h)/3 + k/3, -k, -2 h + k}, {1/2 - h, -((2 h)/3) + k/3, 1 - h + k}, 1/4] == (1/( h k))(h + k) (2 (h - k) + (-4 h + 3 k) HypergeometricPFQ[{1, -2 h, -h, 1 - (4 h)/5 + (3 k)/5, -k, -2 h + k}, {1/2 - h, 1 + h, -((4 h)/5) + (3 k)/5, 1 + k, 1 - h + k}, -(1/4)]); 2 Beta[h, k] HypergeometricPFQ[{-h, -h - k, 1 - (2 h)/3 - (2 k)/3, -2 k}, {1/2 - h/2 - k, 1 - h/2 - k, -((2 h)/3) - (2 k)/3}, 1/4] == (1/( h k))(-h - 2 k + (3 h + 4 k) HypergeometricPFQ[{1, -h, -h - 2 k, -h - k, 1 - (3 h)/5 - (4 k)/5, -2 k}, {1 + h, 1/2 - h/2 - k, 1 - h/2 - k, -((3 h)/5) - (4 k)/5, 1 + k}, -(1/4)]) (Note +1/4 vs -1/4, and additive terms, and upper-parameter 1) I expected them to come out in Gammas (they may yet), but a trivariate polynomial unexpectedly vanished and I had to take a limit. --rwg