For just a moment I thought you'd busted the Curse of Gamma that prevents us from finding a good series for ! that will give a T4 computing time for real! . But it does appear that you've found a polynomial diffeqn for !, which I thought had been proved impossible. Rich PS: Utah-Colorado border has a glitch. I noticed it on a large map & thought "printing error", but a web search turned up the story of a surveying error. The UT-CO glitch is a moderately long segment that isn't quite due North. --R -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of R. William Gosper Sent: Wednesday, March 16, 2005 11:18 AM To: math-fun@mailman.xmission.com Subject: [math-fun] ~s highly miscellaneous hypergeometric identity Just to shake off some rust, I fired up an old Macsyma notebook containing some matrix product rearrangements. A variable dropped out from an unusual spot, producing oo n 5 ==== (n + a) (11 n + 6 a + 1) 4 (n + a - -)! (n + 2 a - 2)! (2 n + a - 1)! \ 6

---------------------------------------------------------------------- / 2 ==== (n + a - -)! (3 n + 2 a + 1)! n = 0 3 1 3 3 (-)! (a - -)! 3 2 = ---------------, 2/3 4 an ungainly 7F6[16/27]. This is a fairly unusual z for a closed form. Also strange is the single a on the right, and the two linear factors in the summand. But oddest is the absence of n!, or (2n)!, or ..., from the summand's denominator, thanks to the dropped variable. Normally, closed forms like this require "termination to the left". So prevalent is that denominator factor that pFq notation supplies it implicitly, yet here it must be artificially (and ungainly) canceled out. A similar but lesser curiosity is oo ==== 2 2 \ 3 k + (3 a + 1) k + a > --------------------------------- = 1, / 2 k + a + 1 ==== (k + a) (k + a + 1) ( ) k = 0 k where the k! is in the numerator instead of the denominator. A slightly uglier cousin gives pi for all a (from rational terms). --rwg Holey mackerel, scombroid cobordism. PPS, the history of the Cal\Nevada border (MKleber> http://laws.findlaw.com/us/447/125.html) is indeed fascinating, The surveying uncertainties dwarfed the mathematical ones. Besides the inevitable problem with longitude, the south end was defined to be where the Colorado River crossed the 35th parallel, which turned out to be highly variable. The town of Aurora, claimed by both states, sent representatives to both legislatures, who simultaneously became house speaker (1862). _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun