I asked:
So who is pushing this twiddle function? Physicists? My mistake. The only one pushing was Wolfram's site search tool, which took me to http://functions.wolfram.com/PDF/Hypergeometric2F1Regularized.pdf instead of the even greater wealth of goodies at http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/ when I searched for cubic transformations. This misled me to think that WRI was trying to abandon traditional 2F1 notation for some inscrutable reason.
Generalizing one of the tabulated special values (07.23.03.0046.01) in the latter compendium, 2 - c (1 - b) (1 - c) hyper_2f1(2, b, c, -----) = ---------------, 1 - b 1 - c + b a telescoping identity. Interestingly, the [c-a,b,c,z/(z-1)] transformation, a 1 - b + a b - 1 hyper_2f1(a, b, a + 2, ---------) = (a + 1) (---------) 1 - b + a 1 - b fails to telescope. Yet hypersimp gets the latter (a lucky beta transformation) but not the former, for failing to try telescopy. (Or summation by parts.) Maybe this evanescent telescopy is related to the matrix version of the z/(z-1) transformation generally producing things messier than 2F1s? --rwg