On 2014-04-02 18:28, Bill Gosper wrote:
On 2014-04-02 14:05, Warren D Smith wrote:
RWG:My objective, more clearly stated, is ultimately to automatically simplify factorials(n/d), not just factorials(n/24), by canonicalizing first to minimal d and then to minimal n. So far, reduction is possible for
n/d, n>d/2 (by reflection) 1/2, 1/6, 3/8, 4/9, n/10, n/12, n/14, n/15, n>1 5/16, 7/16, n/18, n/20, n>1 n/21, n>2 n/22, n/24, n>1 11/25, 12/25,
Add n/26, Add 10/27, 11/27, 13/27, 5/28, 9/28, 11/28, 13/28, n/30,
strongly suggesting n/(4d+2)
n[/] prime, apparently never for prime>2n
{1,7,11,13}/30 use only (1/3)! and ((1/5)! or (2/5)!), e.g., (11/30)! = (11 (1 + Sqrt[5])^(11/60) (5 + Sqrt[5])^(5/12) Sqrt[π] Cos[π/30]^(3/10) Cos[(2 π)/15]^(11/30) (1/5)! (Sec[π/15] Sin[π/30])^(4/15) Sin[(2 π)/15]^(49/60))/ (9 2^(53/60) 3^(1/20) 5^(1/6) Cos[(7 π)/30]^(41/60) (1/3)! Sin[(7 π)/30]^(7/20))
Cos[(7 π)/30]^(41/60) ?
It would be amazing to find a reduction inaccessible via reflection and tuplication. But how could we tell?
WDS> --In fact, Chowla & Selberg *DID* find cases inaccessible by
reflection and tuplication, all (n/24)! can be expressed in close form in terms of elliptic functions. No? There's been followup work by other authors.
Do you know how they demonstrated inaccessibility?
I've posted about this in the past on math-fun, you should check the
archives,
I think the cites are there.
rwg> how could we tell?
wds> wikipedia:
It has been proved that GAMMA(n+r) is a transcendental number and algebraically independent of pi for any integer n and each of the fractions r = 1/6, 1/4, 1/3, 2/3, 3/4, and 5/6.
citing: http://www.math.jussieu.fr/~miw/articles/pdf/TranscendencePeriods.pdf which used the connection to elliptic functions.
more generally, if you had algebraic or rational independence results you "could tell."
Other papers: http://www-fourier.ujf-grenoble.fr/~rivoal/articles/gammater.pdf http://www.mast.queensu.ca/~murty/gun-murty-rath2.pdf in particular note Nesterenko's theorem in the latter. <WDS Wow, I didn't know those transcendences (transcendalities?) had been settled, let alone by the Chudnovskys.
I believe our previous interest in (n/24)! was their rapid computability via Dedekind eta. Even assuming it is always possible to convert to etas, do their modular properties tell all? And, assuming we can canonicalize the eta expression, can we always convert back to Gammas?
I now recall a paper showing the latter. *Large numbers* of Gammas. We ought to be able to prove (n/prime)! inexpressible in terms of (a/prime)!, a<n, or (a/d)!, d<prime, (if prime >2n). And we ought to be able to write (n/(4d+2))! in terms of (a/b)!, b≤2d+1.
I don't think Selberg & Chowla were thinking about computer algebra. I'm just trying to simplify piles of Gammas.
A weird possibility: Allow piles of Beta[rational,rational]. Although traditionally defined in terms of three Gammas, Beta is a much simpler function with nice hypergeometric representations. --rwg Oh, my aching tush: Mathematica is pretty good about Solving linear systems which require a symbolic determinant to vanish. ("Unable to determine if blah=blahblah. Assuming it is ...") But if you append (1/6)! == (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3)) or a similar identity in constants to a vanilla nonsingular system, you instantly get the empty set of solutions!
E.g., In[260]:= Solve[{x == 1, y == 2, (1/6)! == (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3))}, {x, y}] Out[260]= {} Yet, In[261]:= Solve[{x == 1, y == 2, x == (3 Sqrt[3/\[Pi]] ((1/3)!)^2)/(2 2^(1/3))/(1/6)!}, {x, y}] During evaluation of In[261]:= Solve::ztest1: Unable to decide whether numeric quantity 1-(3 Sqrt[3] ((1/3)!)^2)/(2 2^(1/3) Sqrt[\[Pi]] (1/6)!) is equal to zero. Assuming it is. >> Out[261]= {{x -> 1, y -> 2}} --rwg