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 -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of R. William Gosper Sent: Thu 11/24/2005 10:34 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] 2F1(golden ratio) I said There are also similar identities for 2F1(phi^-2). E.g, 3 - sqrt(5) hyper_2f1(a, 1 - 2 a, 3 a, -----------) = 2 %pi 1 4 csc(---) gamma(a + -) gamma(a + -) gamma(3 a) 25 (sqrt(5) + 5) a 5 5 5 (----------------) --------------------------------------------- . 2 4 %pi gamma(5 a)
Once all this goes into the HYPERSIMP facility, we'll see whether 2F1(a,5a,3a,1/phi) or some contiguous neighbor has a monomial rhs.
The z <- z/(z-1) transformation of the above is a nice one: 1 - sqrt(5) hyper_2f1(a, 5 a - 1, 3 a, -----------) = 2 1 - 5 a ------- %pi 2 %pi csc(---) 5 gamma(3 a) 5 ---------------------------------- 2 3 gamma(a) gamma(a + -) gamma(a + -) 5 5 This rhs is a rational multiple of one of the two terms of yesterday's rhs for hyper_2f1(a, 5 a + 1, 3 a + 1, -1/%phi). I'm not sure if there is a single 2F1 equal to a rational multiple of the other term. I'm curious if these are old, since my derivation led through badly swollen intermediate expressions and seemingly overdetermined systems of equations. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun