I said
...(A&S) 15.1.31 seems to be cursed with some sort of singularity that thwarts population of the contiguity grid starting with F(a) and F(a+-3).
Indeed, if you substitute the undefined symbol z for cis(pi/3), the contiguator sez <2F1 you want> = LC<2F1s you know>/(z^2-z+1), i.e., 0/0. When you take the limit, l'Hospital creates <another 2F1 you want> with the same problem! But happily, sometimes you really can, with enough work, create canonical companions from just a single identity. E.g., starting with the [c-a,c-b,c,z] transformation of 5 4 (d140) hyper_2f1(3 a, 5 a - -, 6 a, -----) = 6 3 %phi 3 a 15 a - 3 1 1 3 %phi gamma(a + -) gamma(a + -) 6 2 %phi 1 (--------------------------- - ---------------------------) 7 13 1 19 gamma(a + --) gamma(a + --) gamma(a + --) gamma(a + --) 30 30 30 30 5 a --- - 1/6 2 /5 (c141) dfloat(subst(%pi,a,%)) (d141) 1.43423424711416d+8 = 1.43423424711417d+8 and <same> with a <- a+1, the contiguator was barely able, with a fresh Macsyma, to find 1 4 (d33) hyper_2f1(3 a, a + -, 6 a, -----) = 2 3 %phi 3 a 3 a + 1 1 5 3 %phi gamma(a + -) gamma(a + -) 6 6 ------------------------------------------ 5 a --- 2 3 7 5 gamma(a + --) gamma(a + --) 10 10 (c34) dfloat(subst(%pi,a,%)) (d34) 15.6851338453014d0 = 15.6851338453017d0 and (c60) expand(subst([%,5 = f,9 = t^2,3 = t,t = 3,f = 5],d53)) 1 4 (d60) hyper_2f1(3 a + 1, a + -, 6 a + 2, -----) = 2 3 %phi 3 a + 1 3 a + 2 5 7 3 %phi gamma(a + -) gamma(a + -) 6 6 ---------------------------------------------- 5 a --- + 1 2 9 11 5 gamma(a + --) gamma(a + --) 10 10 (c61) dfloat(subst(%pi,a,%)) (d61) 14.9610163002867d0 = 14.9610163002867d0 I also said
It appears that [Zeilbeger's] search methodology has the advantages of higher automation and lighter algebra,
Maybe not the latter. His "Forty Strange ..." paper said Maple (whose data structures are generally denser than Macsyma's) was only able to explore out to +-4a before exhausting time and memory. But I got the matrices for 6a fairly easily with PC Macsyma, which still thinks it's running on the Intel 286. I also mentioned
http://functions.wolfram.com/PDF/Hypergeometric2F1Regularized.pdf, which contains a wealth of goodies,
Caution: Regularized means divided by Gamma(c), flagged by a twiddle over the F. Ignoring this will lead you to wonderfully bogus Gamma formulas. They seem to pretend that traditional 2F1 notation never existed, but then switch to it (without defining it) occasionally after page 10, where, ironically, it slightly complicates the contiguity formulas. One worth adding (also to A&S): (c87) substpart(factor(piece), contiguate(diff(hyper_2f1(a,b,c,z),z),hyper_2f1(a,b,c,z),hyper_2f1(a,b,c+1,z)),1,1) hyper_2f1(a, b, c, z) (c - b - a) (d87) - --------------------------------- z - 1 hyper_2f1(a, b, c + 1, z) (c - a) (c - b) + ----------------------------------------- c (z - 1) a hyper_2f1(a + 1, b + 1, c + 1, z) b + ------------------------------------- = 0 c relating the d/dz to a canonical pair. Just for convenience. Oops, here it is, 2~F1 = 2F1/Gamma(c), on page 27! And the very last page, 57, gives a url to their notation definitions. Missing: a ToC, and special values (or a pointer to them elsewhere at Wolfram.com). So who is pushing this twiddle function? Physicists? --rwg "Never learn math from a physicist." --Gene Salamin