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May 15th, 2016, 01:56 AM   #1
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gamma(2/3)

gamma(2/3) is transcendental correct?
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May 15th, 2016, 05:42 AM   #2
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Wolfram Alpha and Wikipedia say that we don't know, which I take to mean "probably".
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May 30th, 2016, 09:19 PM   #3
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Well surely there is some means of coming to a consensus; wiki says 1/6 1/2 and 1/3 are transcendental, but you are right the definitions are getting quite muddled but I've done a lot on gamma, which mean quite a long list of functions I know, but when you look at how often gamma pops up in the functional equations for so many others, then as well as the factorial at integer arguments, it's certainly interesting.

But I do have reason to believe it is transcendental and it's looking very promising for me if Liouville's constant is transcendental, which I am starting to see as some kind of "pivot" between transcendental and algebraic irrational numbers of the various degrees, speaking of which, does have a an upper bound right?

Last edited by skipjack; May 30th, 2016 at 11:16 PM.
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May 30th, 2016, 09:24 PM   #4
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but yep again yesterday's article I read had the opinion of not so high thoughts of Liouville's and the general sense I get is a collective uncertainty but then again it's not like anyone is going to stick their neck out on wiki are they. anything id put up would get immediately taken down. Hmmm. That's odd. wtf why would I be certain of something like that? Oh well.

Last edited by skipjack; May 30th, 2016 at 11:17 PM.
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May 30th, 2016, 09:41 PM   #5
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Anyway, I certainly have to agree with the author in wiki that remarked that they are probably transcendental, I mean we know that products and sums of transcendental arguments are themselves also transcendental arguments, and I'm not going to go into any further detail here but I mean clearly at least someone else has gotten to the same conclusions I have somehow, but this one that I posted on another unrelated question is relevant to this discussion (attached):
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File Type: jpg 0 IDENTITY FOR GAMMA.JPG (25.0 KB, 8 views)

Last edited by skipjack; May 30th, 2016 at 11:20 PM.
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May 30th, 2016, 09:46 PM   #6
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you can see that gamma(1/2) and gamma(3/4) of the same type being positioned in this functional root so a rearrangement of the above could then be written to express them as a linear sum, so that ties them together really ie both transcendental or neither it cannot be a mixed pair is my take on it anyway.
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May 30th, 2016, 09:53 PM   #7
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This would then also make the square of gamma(3/4) also transcendental, but it all relies on the proof of gamma(1/2) being transcendental or not transcendental, once we have established that, the collary will then be possible for any rational into gamma ie gamma(n/m)
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May 31st, 2016, 07:05 AM   #8
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Airy Zeta Function -- from Wolfram MathWorld

If you take a look at what this chap Borwein did in 2004 that is on that page, in particular (7) which ive attached, because he has shown the gamma(2/3) and gamma(1/3) can be expressed on opposing sides of the equality operator of a linear equation, in transcendental coefficients/( cofactors whatever you want) thus they are both inherently transcendental, in the same regard as a similar one with as few other fractional gammas in a linear combination . in the one I found the "non gamma" trancendental is pi squared over sqrt(2) and in Borwein's Identity they are 2*pi and 3 ^ (1/6) and 3^(5/6).

Well his arguments good enough for me anyway it works conceptually from what I have been trying to understand further about trancendentals
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May 31st, 2016, 07:09 AM   #9
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ie just get all the gamma expressions on on side and everything non gamma on the other side, and for me that covers a shit ton of expressions that I have studied, the gamma is just literally everywhere it seems.
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May 31st, 2016, 08:52 AM   #10
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Quote:
Originally Posted by Adam Ledger View Post
... algebraic irrational numbers of the various degrees, speaking of which, does have a an upper bound right?
No, there is no upper bound on the degree of algebraic numbers. We can only say that every algebraic number has a degree which is finite.

Quote:
Originally Posted by Adam Ledger View Post
we know that products and sums of transcendental arguments are themselves also transcendental arguments
This is false. $\pi$ and $(4-\pi)$ are both transcendental (because the algebraic numbers are closed under addition) but their sum is not transcendental. $\pi$ and $\frac1\pi$ are both transcendental (because the algebraic numbers are closed under multiplication) but their product is not transcendental.

In general, proving that a number is not algebraic is extremely difficult.

Last edited by v8archie; May 31st, 2016 at 09:15 AM.
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