Is $Gamma{(frac{1}{5})}$ transcendental?
1
$begingroup$
Wolfram Mathworld lists several transcendental numbers such as
$$Gamma{left(frac{1}{3}right)},Gamma{left(frac{1}{4}right)},Gamma{left(frac{1}{6}right)}$$
I don't see the reason why Wolfram Mathworld skips $Gamma{left(frac{1}{5}right)}$.
Is it because it has not yet been proven that $Gamma{left(frac{1}{5}right)}$ is transcendental? If so, what makes it hard to prove the transcendentality of $Gamma{left(frac{1}{5}right)}$?
gamma-function transcendental-numbers
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
$endgroup$
|
show 1 more comment
1
$begingroup$
Wolfram Mathworld lists several transcendental numbers such as
$$Gamma{left(frac{1}{3}right)},Gamma{left(frac{1}{4}right)},Gamma{left(frac{1}{6}right)}$$
I don't see the reason why Wolfram Mathworld skips $Gamma{left(frac{1}{5}right)}$.
Is it because it has not yet been proven that $Gamma{left(frac{1}{5}right)}$ is transcendental? If so, what makes it hard to prove the transcendentality of $Gamma{left(frac{1}{5}right)}$?
gamma-function transcendental-numbers
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
$endgroup$
2
$begingroup$
I expect it is unknown. this suggests that transcendence is known for very few values.
$endgroup$
– lulu
Dec 19 '18 at 1:23
$begingroup$
So we can't approximate it using quadratically convergent arithmetic-geometric mean. Is it the reason why it is difficult?
$endgroup$
– Larry
Dec 19 '18 at 1:31
$begingroup$
I suspect the answer to this is very technical - it doesn't look like it was straightforward to show that the numbers you list are transcendental, and it could either be that the methods used were just inherently focused on one value or that there's some subtle problem in the techniques that prevents them from generalizing at all. In either case, it'd likely be necessary to understand why those numbers are transcendental in order to understand why it's hard to show that for $Gamma(1/5)$.
$endgroup$
– Milo Brandt
Dec 19 '18 at 1:31
$begingroup$
@Brandt: Helpful links or general explanation would be fine. I am not necessarily looking for a technical answer (it would be great if someone provides a detailed explanation).
$endgroup$
– Larry
Dec 19 '18 at 1:39
1
$begingroup$
Founds this - math.stackexchange.com/questions/2360504/…
$endgroup$
– user150203
Jan 2 at 4:43
|
show 1 more comment
1
1
1
$begingroup$
Wolfram Mathworld lists several transcendental numbers such as
$$Gamma{left(frac{1}{3}right)},Gamma{left(frac{1}{4}right)},Gamma{left(frac{1}{6}right)}$$
I don't see the reason why Wolfram Mathworld skips $Gamma{left(frac{1}{5}right)}$.
Is it because it has not yet been proven that $Gamma{left(frac{1}{5}right)}$ is transcendental? If so, what makes it hard to prove the transcendentality of $Gamma{left(frac{1}{5}right)}$?
gamma-function transcendental-numbers
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
$endgroup$
Wolfram Mathworld lists several transcendental numbers such as
$$Gamma{left(frac{1}{3}right)},Gamma{left(frac{1}{4}right)},Gamma{left(frac{1}{6}right)}$$
I don't see the reason why Wolfram Mathworld skips $Gamma{left(frac{1}{5}right)}$.
Is it because it has not yet been proven that $Gamma{left(frac{1}{5}right)}$ is transcendental? If so, what makes it hard to prove the transcendentality of $Gamma{left(frac{1}{5}right)}$?
gamma-function transcendental-numbers
gamma-function transcendental-numbers
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
edited Dec 19 '18 at 11:39
Larry
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
asked Dec 19 '18 at 1:20
LarryLarry
2,53031131
2,53031131
2
$begingroup$
I expect it is unknown. this suggests that transcendence is known for very few values.
$endgroup$
– lulu
Dec 19 '18 at 1:23
$begingroup$
So we can't approximate it using quadratically convergent arithmetic-geometric mean. Is it the reason why it is difficult?
$endgroup$
– Larry
Dec 19 '18 at 1:31
$begingroup$
I suspect the answer to this is very technical - it doesn't look like it was straightforward to show that the numbers you list are transcendental, and it could either be that the methods used were just inherently focused on one value or that there's some subtle problem in the techniques that prevents them from generalizing at all. In either case, it'd likely be necessary to understand why those numbers are transcendental in order to understand why it's hard to show that for $Gamma(1/5)$.
$endgroup$
– Milo Brandt
Dec 19 '18 at 1:31
$begingroup$
@Brandt: Helpful links or general explanation would be fine. I am not necessarily looking for a technical answer (it would be great if someone provides a detailed explanation).
$endgroup$
– Larry
Dec 19 '18 at 1:39
1
$begingroup$
Founds this - math.stackexchange.com/questions/2360504/…
$endgroup$
– user150203
Jan 2 at 4:43
|
show 1 more comment
2
$begingroup$
I expect it is unknown. this suggests that transcendence is known for very few values.
$endgroup$
– lulu
Dec 19 '18 at 1:23
$begingroup$
So we can't approximate it using quadratically convergent arithmetic-geometric mean. Is it the reason why it is difficult?
$endgroup$
– Larry
Dec 19 '18 at 1:31
$begingroup$
I suspect the answer to this is very technical - it doesn't look like it was straightforward to show that the numbers you list are transcendental, and it could either be that the methods used were just inherently focused on one value or that there's some subtle problem in the techniques that prevents them from generalizing at all. In either case, it'd likely be necessary to understand why those numbers are transcendental in order to understand why it's hard to show that for $Gamma(1/5)$.
$endgroup$
– Milo Brandt
Dec 19 '18 at 1:31
$begingroup$
@Brandt: Helpful links or general explanation would be fine. I am not necessarily looking for a technical answer (it would be great if someone provides a detailed explanation).
$endgroup$
– Larry
Dec 19 '18 at 1:39
1
$begingroup$
Founds this - math.stackexchange.com/questions/2360504/…
$endgroup$
– user150203
Jan 2 at 4:43
2
2
$begingroup$
I expect it is unknown. this suggests that transcendence is known for very few values.
$endgroup$
– lulu
Dec 19 '18 at 1:23
$begingroup$
I expect it is unknown. this suggests that transcendence is known for very few values.
$endgroup$
– lulu
Dec 19 '18 at 1:23
$begingroup$
So we can't approximate it using quadratically convergent arithmetic-geometric mean. Is it the reason why it is difficult?
$endgroup$
– Larry
Dec 19 '18 at 1:31
$begingroup$
So we can't approximate it using quadratically convergent arithmetic-geometric mean. Is it the reason why it is difficult?
$endgroup$
– Larry
Dec 19 '18 at 1:31
$begingroup$
I suspect the answer to this is very technical - it doesn't look like it was straightforward to show that the numbers you list are transcendental, and it could either be that the methods used were just inherently focused on one value or that there's some subtle problem in the techniques that prevents them from generalizing at all. In either case, it'd likely be necessary to understand why those numbers are transcendental in order to understand why it's hard to show that for $Gamma(1/5)$.
$endgroup$
– Milo Brandt
Dec 19 '18 at 1:31
$begingroup$
I suspect the answer to this is very technical - it doesn't look like it was straightforward to show that the numbers you list are transcendental, and it could either be that the methods used were just inherently focused on one value or that there's some subtle problem in the techniques that prevents them from generalizing at all. In either case, it'd likely be necessary to understand why those numbers are transcendental in order to understand why it's hard to show that for $Gamma(1/5)$.
$endgroup$
– Milo Brandt
Dec 19 '18 at 1:31
$begingroup$
@Brandt: Helpful links or general explanation would be fine. I am not necessarily looking for a technical answer (it would be great if someone provides a detailed explanation).
$endgroup$
– Larry
Dec 19 '18 at 1:39
$begingroup$
@Brandt: Helpful links or general explanation would be fine. I am not necessarily looking for a technical answer (it would be great if someone provides a detailed explanation).
$endgroup$
– Larry
Dec 19 '18 at 1:39
1
1
$begingroup$
Founds this - math.stackexchange.com/questions/2360504/…
$endgroup$
– user150203
Jan 2 at 4:43
$begingroup$
Founds this - math.stackexchange.com/questions/2360504/…
$endgroup$
– user150203
Jan 2 at 4:43
|
show 1 more comment
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2
$begingroup$
I expect it is unknown. this suggests that transcendence is known for very few values.
$endgroup$
– lulu
Dec 19 '18 at 1:23
$begingroup$
So we can't approximate it using quadratically convergent arithmetic-geometric mean. Is it the reason why it is difficult?
$endgroup$
– Larry
Dec 19 '18 at 1:31
$begingroup$
I suspect the answer to this is very technical - it doesn't look like it was straightforward to show that the numbers you list are transcendental, and it could either be that the methods used were just inherently focused on one value or that there's some subtle problem in the techniques that prevents them from generalizing at all. In either case, it'd likely be necessary to understand why those numbers are transcendental in order to understand why it's hard to show that for $Gamma(1/5)$.
$endgroup$
– Milo Brandt
Dec 19 '18 at 1:31
$begingroup$
@Brandt: Helpful links or general explanation would be fine. I am not necessarily looking for a technical answer (it would be great if someone provides a detailed explanation).
$endgroup$
– Larry
Dec 19 '18 at 1:39
1
$begingroup$
Founds this - math.stackexchange.com/questions/2360504/…
$endgroup$
– user150203
Jan 2 at 4:43