[math-fun] 2F1(golden ratio)
I have the weird feeling I once knew this and forgot it, but since I no longer have a decent message massager (or "masausage", as a young friend recently misspoke) we'll have to risk repetition. 1 - sqrt(5) hyper_2f1(a, 5 a + 1, 3 a + 1, -----------) = 2 5 a --- 2 3 2 pi 5 (a - -)! (a - -)! 4 %pi cos(--) 5 5 10 (3 a)! (---------------------- + ----------------------------) pi 5 a (5 a)! 4 %pi cos(--) --- 10 2 3 2 5 a! 5 (a - -)! (a - -)! 5 5 Note G/(5a)! + 1/G/5a!. I think this is almost, but not quite derivable from the formulas in A&S Chapter 15, starting with 15.1.31, and then various linear and quadratic transformations. The obstacle seems to be A&S's negligence to provide contiguous pairs both for 15.1.31 and the quadratic transformations. The latter can be somewhat tediously derived from the three-term recurrences, but 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). (Of course, the good way to do things is with 3x3 matrices that have the contiguous pair built in. I'm belatedly nagging the DLMF people about this.) There are also similar identities for 2F1(phi^-2). 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. --rwg
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
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
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}. Raimundas Vid\={u}nas: {Darboux evaluations of algebraic Gauss hypergeometric functions}, arXiv:math.CA/0504264, 13-April-2005. Online at \url{http://arxiv.org/abs/math/0504264}. For the DLMF these might be fine input: Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, 30-March-2004. Online at \url{http://arxiv.org/abs/math/0403510}. And pretty please Garvan's two papers giving AGM iterations for F([1/2+s,1/2-s],[1],z) (and some others): J.\ Borwein, P.\ Borwein, Frank Garvan: {Hypergeometric Analogues of the Arithmetic-Geometric Mean Iteration}, Constr.\ Approx.\ 9, 1993.} %Online at \url{http://www.math.ufl.edu/fac/facmr/Garvan.html}. (disappeared) Frank Garvan: {Cubic modular identities of Ramanujan, hypergeometric functions \ and analogues of the arithmetic-geometric mean iteration}, Contemporary Mathematics 166, 1993. %Online at \url{http://www.math.ufl.edu/fac/facmr/Garvan.html}. (disappeared) * R. William Gosper <rwg@osots.com> [Nov 25. 2005 06:37]:
I said There are also similar identities for 2F1(phi^-2).
[...]
-- p=2^q-1 prime <== q>2, cosh(2^(q-2)*log(2+sqrt(3)))%p=0 Life is hard and then you die.
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
the book is here : http://www.lacim.uqam.ca/%7Eplouffe/articles/Abramowitz&Stegun.pdf OCR has been made on it too. Simon plouffe
--- Jud McCranie <j.mccranie@adelphia.net> wrote:
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.
I just noticed this. When and where is Abramowitz and Stegun to be online?
http://dlmf.nist.gov/ Gene __________________________________ Yahoo! Mail - PC Magazine Editors' Choice 2005 http://mail.yahoo.com
At 02:55 PM 11/27/2005, you wrote:
I just noticed this. When and where is Abramowitz and Stegun to be
online?
Thanks!
Simon Plouffe has a scanned copy of A&S online at http://www.lacim.uqam.ca/~plouffe/articles/Abramowitz&Stegun.pdf It's huge, almost 1000 pages of images. Very handy to have a local copy downloaded. Rich ________________________________ From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Jud McCranie Sent: Sun 11/27/2005 9:58 AM To: math-fun Subject: [math-fun] Abramowitz & Stegun
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.
I just noticed this. When and where is Abramowitz and Stegun to be online? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Dim, nov 27, 2005 at 01:08:25 -0700, Schroeppel, Richard wrote:
Simon Plouffe has a scanned copy of A&S online at
http://www.lacim.uqam.ca/~plouffe/articles/Abramowitz&Stegun.pdf
And I once found a DJVU version online (much smaller file than PDF and much quicker navigation with djview than with xpdf). If I have the right to do it, I can put it somewhere, but would it be legal ? -- Thomas Baruchel Home Page: http://baruchel.free.fr/~thomas/
On Lun, nov 28, 2005 at 11:10:09 +0100, Thomas Baruchel wrote:
On Dim, nov 27, 2005 at 01:08:25 -0700, Schroeppel, Richard wrote:
Simon Plouffe has a scanned copy of A&S online at
http://www.lacim.uqam.ca/~plouffe/articles/Abramowitz&Stegun.pdf
And I once found a DJVU version online (much smaller file than PDF and much quicker navigation with djview than with xpdf).
If I have the right to do it, I can put it somewhere, but would it be legal ?
I have my own DJVU version but I also found another version today : http://lib.org.by./_djvu/M_Mathematics/MRef_References/ (only working with a proxy in Poland or in BielloRussia like 195.116.60.65 (port: 3127) Of course you can download it as long as it is in the public domain, but you very certainly are not allowed to download the very many other books in http://lib.org.by./_djvu -- Thomas Baruchel Home Page: http://baruchel.free.fr/~thomas/
http://mintaka.sdsu.edu/faculty/wfw/ABRAMOWITZ-STEGUN/ Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com
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
I asked:
So who is pushing this twiddle function? Physicists? My mistake. The only one pushing was Wolfram's site search tool, which took me to http://functions.wolfram.com/PDF/Hypergeometric2F1Regularized.pdf instead of the even greater wealth of goodies at http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/ when I searched for cubic transformations. This misled me to think that WRI was trying to abandon traditional 2F1 notation for some inscrutable reason.
Generalizing one of the tabulated special values (07.23.03.0046.01) in the latter compendium, 2 - c (1 - b) (1 - c) hyper_2f1(2, b, c, -----) = ---------------, 1 - b 1 - c + b a telescoping identity. Interestingly, the [c-a,b,c,z/(z-1)] transformation, a 1 - b + a b - 1 hyper_2f1(a, b, a + 2, ---------) = (a + 1) (---------) 1 - b + a 1 - b fails to telescope. Yet hypersimp gets the latter (a lucky beta transformation) but not the former, for failing to try telescopy. (Or summation by parts.) Maybe this evanescent telescopy is related to the matrix version of the z/(z-1) transformation generally producing things messier than 2F1s? --rwg
participants (7)
-
Eugene Salamin -
Joerg Arndt -
Jud McCranie -
R. William Gosper -
Schroeppel, Richard -
Simon Plouffe -
Thomas Baruchel