Page xii of Basic Hypergeometric Series says that Pfaff's 3F2[1] (long misattributed to Saalschütz) preceded Gauss's 2F1[1] limiting case by 16 years! (Did Pfaff ever think to take the n→∞ limit? He must have. But then why didn't he notice that the 2F1 was more useful? He was Gauss's research supervisor and housemate(?). Maybe he deliberately gave Gauss the credit.) Yikes, he also discovered Hypergeometric2F1[a_, b_, c_, z_] :> Hypergeometric2F1[c - a, b, c, z/(z - 1)]/(1 - z)^b ! --rwg Mathematica 10.3 still fails Pfaff: In[613]:= Assuming[n \[Element] Integers && n >= 0, FunctionExpand[HypergeometricPFQ[{a, b, -n}, {d, a + b - n - d + 1}, 1]]] Out[613]= HypergeometricPFQ[{a, b, -n}, {d, 1 + a + b - d - n}, 1] Reminder: If you don't know which parameter is the nonpositive integer, (c67) hypersimp(hyper_f[3,2]([a,b,c],[d,a+b+c-d+1])) Is any of a, b, or c a nonpositive integer? yes; (d67) - ((%pi)^2 * Gamma( - d + c + b + a + 1) * Gamma(d)/(Gamma( - d + b + a + 1) * Gamma( - d + c + a + 1) * Gamma( - d + c + b + 1) * Gamma(d - a) * Gamma(d - b) * Gamma(d - c) * (cos(%pi * d) * cos(%pi * (d - c - b - a)) - cos(%pi * a) * cos(%pi * b) * cos(%pi * c)))) (c68) neg_a_part(%,1,2,-1) (d68) ((%pi)^2 * Gamma( - d + c + b + a + 1) * Gamma(d)/(Gamma( - d + b + a + 1) * Gamma( - d + c + a + 1) * Gamma( - d + c + b + 1) * Gamma(d - a) * Gamma(d - b) * Gamma(d - c) * (cos(%pi * a) * cos(%pi * b) * cos(%pi * c) - cos(%pi * d) * cos(%pi * (d - c - b - a))))) (Also works: standardize_signs(%,d). Boy, does Mathematica ever need neg_a_part and standardize_signs.)