rcs>Will the digital version of Abramowitz & Stegun include a compendium of results like this? The old A&S has a couple of dilog values, but it's pretty thin. Rich I'm bcc'ing Askey. There's a fair selection ~ 15.1.20, but nowehere near as complete as I once thought. But with all the applicable linear, quadratic and even cubic transformations, there may be too many pairs(!) to tabulate. Maybe they could choose a set of representative pairs independent under the transformations, each accompanying a list of all the [a,b;c|z] tuples reachable via the transformations. Then, a computer, at least, could follow the trail to the actual identity, provided that the quadratic and cubic transformations are generalized to map pairs to pairs. Joerg Arndt> Just in case you haven't seen: Shalosh B.EKHAD (Doron Zeilberger): {Forty Strange Computer-Discovered Hypergeometric Series Evaluations}, 12-October-2004. Online at \url{http://www.math.rutgers.edu/~zeilberg/pj.html}. Thank you, I didn't know about that! His Theorem 34 is the special (terminating) case a = negative integer of "my" "recent" 3 a 3 a - 1 5 1 3 %phi (a - -)! (a - -)! 1 6 6 hyper_2f1(2 a, 1 - 2 a, 3 a + -, 1 - %phi) = ---------------------------------- 2 5 a --- 2 7 3 5 (a - --)! (a - --)! 10 10 (hyper_2f1(2*a,1-2*a,3*a+1/2,1-%phi) = 3^(3*a)*%phi^(3*a-1)*(a-5/6)!*(a-1/6)!/(5^(5*a/2)*(a-7/10)!*(a-3/10)!) ) for complex a. It appears that his search methodology has the advantages of higher automation and lighter algebra, and the disadvantages of restriction to integer parameters and monomial RHSs. The latter would preclude discovery of the highly desirable contiguous companion 1 hyper_2f1(2 a, 1 - 2 a, 3 a - -, 1 - %phi) = 2 1 - 5 a ------- 2 3 a - 2 3 a - 1 1 1 5 %phi 3 gamma(a - -) gamma(a + -) 6 6 1 %phi (--------------------------- + ---------------------------), 3 3 1 1 gamma(a - --) gamma(a + --) gamma(a - --) gamma(a + --) 10 10 10 10 ( hyper_2f1(2*a,1-2*a,3*a-1/2,1-%phi) = 5^((1-5*a)/2)*%phi^(3*a-2)*3^(3*a-1)*gamma(a-1/6)*gamma(a+1/6)* (1/(gamma(a-3/10)*gamma(a+3/10))+%phi/(gamma(a-1/10)*gamma(a+1/10)))) which leads to an infinite, three dimensional grid of related identities. See sample Macsyma output at the end of this message. Note the tricky pattern-match [2*a,1-2*a; 3*a-1/2] = (mod 1) [b, -b; -3b/2] because the substitution a <- a+1/2 is contiguous to the monomial identity (strange to find so many after so long) hyper_2f1(2 a, 1 - 2 a, 3 a, 1 - %phi) = 5 a --- %pi 2 3 a - 1 1 4 csc(---) 5 %phi gamma(a + -) gamma(a + -) gamma(3 a) 5 5 5 --------------------------------------------------------------, 4 %pi gamma(5 a) (hyper_2f1(2*a,1-2*a,3*a,1-%phi) = csc(%pi/5)*5^(5*a/2)*%phi^(3*a-1)*gamma(a+1/5) *gamma(a+4/5)*gamma(3*a)/(4*%pi*gamma(5*a)) ) effectively doubling the density of the "c" (lower parameter) axis of the contiguity grid. It seems to me Zeilberger's robot should have found this as a significant, distinct case. In any case, the "Zeilberger 40" are valuable indicators of the existence and form of their generalizations to complex and contiguous parameters. Who wants to pay me to chase them all down?-) It will be particularly interesting to see how many are truly "strange", and how many are consequences of known transformations of known identities. Besides Joerg's helpful URLs, I just found http://functions.wolfram.com/PDF/Hypergeometric2F1Regularized.pdf, which contains a wealth of goodies, including my first look at the cubic transformations. It looks like Wolfram is giving DLMF a run for its money. --rwg (c185) (hyper_2f1(2*a,1-2*a,3*a+1/2,1-%phi),%% = hypersimp(%%)) 1 (d185) hyper_2f1(2 a, 1 - 2 a, 3 a + -, 1 - %phi) = 2 3 a - 2 3 a - 2 1 5 9 (sqrt(5) + 1) 3 %phi gamma(a + -) gamma(a + -) 6 6 -------------------------------------------------------------- 5 a --- 2 3 7 2 5 gamma(a + --) gamma(a + --) 10 10 (c186) dfloat(subst(%pi,a,%)) (d186) 5.99118801081389d0 = 5.99118801081391d0 (c187) (hyper_2f1(2*a,1-2*a,3*a,1-%phi),%% = hypersimp(%%)) (d187) hyper_2f1(2 a, 1 - 2 a, 3 a, 1 - %phi) = 6 a - 7 6 a - 7 ------- ------- + 1 2 2 1 2 54 3 %phi gamma(a + -) gamma(a + -) 3 3 - -------------------------------------------------------- 10 a - 7 -------- - 1 2 ------------ 2 2 3 25 (sqrt(5) - 3) 5 gamma(a + -) gamma(a + -) 5 5 (c188) dfloat(subst(%pi,a,%)) (d188) 6.46924356533372d0 = 6.46924356533372d0 (c189) (hyper_2f1(b,-b,-3*b/2,1/(-%phi)),%% = hypersimp(%%)) 3 b 1 (d189) hyper_2f1(b, - b, - ---, - ----) = 2 %phi 1 b 2 b 5 (sqrt(5) + 2) 2 gamma(- - -) gamma(- - -) (------------------------------------------ 3 2 3 2 1 b 4 b (7 sqrt(5) + 15) gamma(- - -) gamma(- - -) 5 2 5 2 1 5 b - 3 (2 sqrt(5) + 5) (---- + 1) ------- %phi 4 + ---------------------------------------) 5 2 b 3 b (sqrt(5) + 3) gamma(- - -) gamma(- - -) 5 2 5 2 3 b + 1 3 b + 1 ------- ------- 1 2 2 /((---- + 1) 3 ) %phi (c190) dfloat(subst(%pi,b,%)) (d190) 0.24996719342713d0 = 0.24996719342739d0 (c157) (hyper_2f1(2*a,1-2*a,3*a-1/2,1-%phi),%% = hypersimp(%%)) 1 (d157) hyper_2f1(2 a, 1 - 2 a, 3 a - -, 1 - %phi) = 2 3 a - 2 3 a - 2 1 1 3 3 %phi gamma(a - -) gamma(a + -) 6 6 5 a - 1 ------- 1 %phi 2 (--------------------------- + ---------------------------)/5 3 3 1 1 gamma(a - --) gamma(a + --) gamma(a - --) gamma(a + --) 10 10 10 10 (c158) dfloat(subst(%pi,a,%)) (d158) 7.0315378879754d0 = 7.03153788797541d0