[math-fun] about the polygamma function
Hello, the Psi function is one of my favorite function, it has so many aspects and formulas, very interesting, for example, you can express primes with it, 691 = Psi(11,1/2)/(Pi^12*2^8). or Euler numbers, (absolute value), I hate that - sign all over. ;-). E(2n) = -1/2*(Psi(k,1/8)-Psi(k,3/8)+Psi(k,5/8)-Psi(k,7/8))*Pi^(-k-1)*4^(-k). we can also use Psi(n,1/4) and Psi(n,3/4) for the same. another example is 67 = 1/64*(Psi(5,1/5)-Psi(5,2/5)-Psi(5,3/5)+Psi(5,4/5))*5^(1/2)/Pi^6 a very good question is : are there other primes like that ? I have such a formula for 17, 31, 41, 61, 67, ...this type of numbers include prime euler numbers and (with some arrangement with the denominator) prime bernoulli numbers as well. Another direction with that idea , a fact known to Ramanujan was that (for example) 24 * sum(n^673/(exp(2*Pi*n)-1),n=1..infinity) = 156344681616153723364... 465357092320036059151 a prime of 1077 digits, that number is in fact close to a Bernoulli number in disguise, B(674) we can express that infinite series with the polygamma function if we like instead. also a fact which I find quite interesting, we have here an expression for some primes which is made with a sum of irrational numbers. other representations are with binomial sums which are simplified if we use the polygamma function with a negative argument, , integer relations with known constants. you may look at this document for more formulas here : http://plouffe.fr/The%20many%20faces%20of%20the%20polygamma%20function.pdf one interesting finding is that I have a polygamma expression for Li1(1/2), Li2(1/2), Li3(1/2), which are <the same> in term of polygamma function, i could not find yet an expression for Li4(1/2), but I am working on it since the first 3 values are very similar. see the formulas on page 3. all formulas are with that same polygamma function, and yes : one last point, the one I use is the one implemented in Maple and not mathematica mainly because the mathematica program uses the wrong (in my opinion) generalization of polygamma with negative arguments, the correct one is the one of Espinosa and Moll. (see the references). bonne lecture, Simon plouffe
Hogamous, higamous, Plouffe is polygammous. Higamous, hogamous, Gosper monogammous. WFL On 4/1/16, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello,
the Psi function is one of my favorite function, it has so many aspects and formulas, very interesting,
for example, you can express primes with it,
691 = Psi(11,1/2)/(Pi^12*2^8).
or Euler numbers, (absolute value), I hate that - sign all over. ;-).
E(2n) = -1/2*(Psi(k,1/8)-Psi(k,3/8)+Psi(k,5/8)-Psi(k,7/8))*Pi^(-k-1)*4^(-k).
we can also use Psi(n,1/4) and Psi(n,3/4) for the same.
another example is 67 =
1/64*(Psi(5,1/5)-Psi(5,2/5)-Psi(5,3/5)+Psi(5,4/5))*5^(1/2)/Pi^6
a very good question is : are there other primes like that ?
I have such a formula for 17, 31, 41, 61, 67, ...this type of numbers include prime euler numbers and (with some arrangement with the denominator) prime bernoulli numbers as well.
Another direction with that idea , a fact known to Ramanujan was that (for example)
24 * sum(n^673/(exp(2*Pi*n)-1),n=1..infinity) = 156344681616153723364... 465357092320036059151 a prime of 1077 digits, that number is in fact close to a Bernoulli number in disguise, B(674) we can express that infinite series with the polygamma function if we like instead.
also a fact which I find quite interesting, we have here an expression for some primes which is made with a sum of irrational numbers.
other representations are with binomial sums which are simplified if we use the polygamma function with a negative argument, , integer relations with known constants.
you may look at this document for more formulas here :
http://plouffe.fr/The%20many%20faces%20of%20the%20polygamma%20function.pdf
one interesting finding is that I have a polygamma expression for Li1(1/2), Li2(1/2), Li3(1/2), which are <the same> in term of polygamma function, i could not find yet an expression for Li4(1/2), but I am working on it since the first 3 values are very similar. see the formulas on page 3.
all formulas are with that same polygamma function, and yes : one last point, the one I use is the one implemented in Maple and not mathematica mainly because the mathematica program uses the wrong (in my opinion) generalization of polygamma with negative arguments, the correct one is the one of Espinosa and Moll. (see the references).
bonne lecture,
Simon plouffe
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hello, I made a mistake, infamous, E(2k) = -1/2*(Psi(k,1/8)-Psi(k,3/8)+Psi(k,5/8)-Psi(k,7/8))*Pi^(-k-1)*4^(-k) I used E(2n) , shame on me, n on one side and k on the other, it is a monogamous error :-). Best regards, Simon Plouffe 2016-04-01 17:12 GMT+02:00 Fred Lunnon <fred.lunnon@gmail.com>:
Hogamous, higamous, Plouffe is polygammous. Higamous, hogamous, Gosper monogammous.
WFL
On 4/1/16, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello,
the Psi function is one of my favorite function, it has so many aspects and formulas, very interesting,
for example, you can express primes with it,
691 = Psi(11,1/2)/(Pi^12*2^8).
or Euler numbers, (absolute value), I hate that - sign all over. ;-).
E(2n) = -1/2*(Psi(k,1/8)-Psi(k,3/8)+Psi(k,5/8)-Psi(k,7/8))*Pi^(-k-1)*4^(-k).
we can also use Psi(n,1/4) and Psi(n,3/4) for the same.
another example is 67 =
1/64*(Psi(5,1/5)-Psi(5,2/5)-Psi(5,3/5)+Psi(5,4/5))*5^(1/2)/Pi^6
a very good question is : are there other primes like that ?
I have such a formula for 17, 31, 41, 61, 67, ...this type of numbers include prime euler numbers and (with some arrangement with the denominator) prime bernoulli numbers as well.
Another direction with that idea , a fact known to Ramanujan was that (for example)
24 * sum(n^673/(exp(2*Pi*n)-1),n=1..infinity) = 156344681616153723364... 465357092320036059151 a prime of 1077 digits, that number is in fact close to a Bernoulli number in disguise, B(674) we can express that infinite series with the polygamma function if we like instead.
also a fact which I find quite interesting, we have here an expression for some primes which is made with a sum of irrational numbers.
other representations are with binomial sums which are simplified if we use the polygamma function with a negative argument, , integer relations with known constants.
you may look at this document for more formulas here :
http://plouffe.fr/The%20many%20faces%20of%20the%20polygamma%20function.pdf
one interesting finding is that I have a polygamma expression for Li1(1/2), Li2(1/2), Li3(1/2), which are <the same> in term of polygamma function, i could not find yet an expression for Li4(1/2), but I am working on it since the first 3 values are very similar. see the formulas on page 3.
all formulas are with that same polygamma function, and yes : one last point, the one I use is the one implemented in Maple and not mathematica mainly because the mathematica program uses the wrong (in my opinion) generalization of polygamma with negative arguments, the correct one is the one of Espinosa and Moll. (see the references).
bonne lecture,
Simon plouffe
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Btw, I was certain that this is a takeoff on a rhyme *written by Aldous Huxley*, having long ago read some non-fiction by or about him that stated he made this up as a schoolboy. So googled to confirm this. But the Internet mainly thinks the origin of this is undetermined (just google on hogamous, higamous). After probably close to 10 minutes of googling I found *one* hit where someone said it was by Aldous Huxley — the others didn't even mention him as a candidate. And that hit was just a comment by someone in a discussion group, no citation. Do you, Fred, or anyone else, know the origin of Hogamous, higamous, Man is polygamous. Higamous, hogamous, Woman monogamous. ? —Dan
On Apr 1, 2016, at 8:23 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hogamous, higamous, Plouffe is polygammous. Higamous, hogamous, Gosper monogammous.
See http://quoteinvestigator.com/2012/03/28/hogamous/ . Jim Propp On Fri, Apr 1, 2016 at 3:52 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Btw, I was certain that this is a takeoff on a rhyme *written by Aldous Huxley*, having long ago read some non-fiction by or about him that stated he made this up as a schoolboy. So googled to confirm this.
But the Internet mainly thinks the origin of this is undetermined (just google on hogamous, higamous). After probably close to 10 minutes of googling I found *one* hit where someone said it was by Aldous Huxley — the others didn't even mention him as a candidate.
And that hit was just a comment by someone in a discussion group, no citation.
Do you, Fred, or anyone else, know the origin of
Hogamous, higamous, Man is polygamous. Higamous, hogamous, Woman monogamous.
?
—Dan
On Apr 1, 2016, at 8:23 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hogamous, higamous, Plouffe is polygammous. Higamous, hogamous, Gosper monogammous.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes, Jim, I read that and many more equally inconclusive web pages. —Dan
On Apr 1, 2016, at 1:25 PM, James Propp <jamespropp@gmail.com> wrote:
http://quoteinvestigator.com/2012/03/28/hogamous/ <http://quoteinvestigator.com/2012/03/28/hogamous/>
The poem can unquestionably be credited to Claire MacMurray who penned it for the Cleveland Plain Dealer in November 1939. No earlier appearance is known and until such is found, there it rests. But wait (I hear you think), she attributed it to a Mrs. Amos Pinchot. Quote Investigator states that "a cite in 1942 claimed that she [Gertrude (Minturn) Pinchot] denied the attribution" and that (therefore, presumably) "no decisive candidate for authorship has yet emerged". Ouch! Gertrude died in May 1939. It would have been difficult for her to deny the November 1939 attribution.
On Apr 1, 2016, at 4:28 PM, Dan Asimov <asimov@msri.org> wrote:
Yes, Jim, I read that and many more equally inconclusive web pages.
On Apr 1, 2016, at 1:25 PM, James Propp <jamespropp@gmail.com> wrote:
Hans, maybe your "unquestionably" is written with glossa firmly in bucca. But of course all we know is that whoever wrote that "Quote Investigator" piece about it was unaware of any mention of that rhyme prior to 1939. This is not the same as knowing there isn't one. (E.g., in 1772 Euler proved the primality of the largest prime known by then: 2,147,483,647. (In 1951, before computers were used for this task, the largest known prime was 20,988,936,657,440,586,486,151,264,256,610,222,593,863,921: 44 decimal digits. (By April 2016, the largest known prime has 22,338,618 decimal digits.) And of course your discovery of when Mrs. Pinchot died brings more doubt about the reliability of Quote Investigator. —Dan
On Apr 1, 2016, at 2:49 PM, Hans Havermann <gladhobo@teksavvy.com> wrote:
The poem can unquestionably be credited to Claire MacMurray who penned it for the Cleveland Plain Dealer in November 1939. No earlier appearance is known and until such is found, there it rests. But wait (I hear you think), she attributed it to a Mrs. Amos Pinchot. Quote Investigator states that "a cite in 1942 claimed that she [Gertrude (Minturn) Pinchot] denied the attribution" and that (therefore, presumably) "no decisive candidate for authorship has yet emerged". Ouch! Gertrude died in May 1939. It would have been difficult for her to deny the November 1939 attribution.
On Apr 1, 2016, at 4:28 PM, Dan Asimov <asimov@msri.org> wrote:
Yes, Jim, I read that and many more equally inconclusive web pages.
On Apr 1, 2016, at 1:25 PM, James Propp <jamespropp@gmail.com> wrote:
Dan: "And of course your discovery of when Mrs. Pinchot died brings more doubt about the reliability of Quote Investigator." To be fair to Garson O'Toole (the Quote Investigator), there had been (since 1919) a *second* Mrs. Pinchot. But if that person denied authorship of the poem I'm certain that Claire MacMurray must have meant the *first*. ;)
Thanks, Hans. —Dan
On Apr 1, 2016, at 5:33 PM, Hans Havermann <gladhobo@teksavvy.com> wrote:
Dan: "And of course your discovery of when Mrs. Pinchot died brings more doubt about the reliability of Quote Investigator." To be fair to Garson O'Toole (the Quote Investigator), there had been (since 1919) a *second* Mrs. Pinchot. But if that person denied authorship of the poem I'm certain that Claire MacMurray must have meant the *first*. ;).
I've summarized my general unease about the hogamus/higamus attributions/misattributions in my blog: http://gladhoboexpress.blogspot.ca/2016/04/she-said-he-said.html
On Fri, 1 Apr 2016, Hans Havermann wrote:
Dan: "And of course your discovery of when Mrs. Pinchot died brings more doubt about the reliability of Quote Investigator." To be fair to Garson O'Toole (the Quote Investigator), there had been (since 1919) a *second* Mrs. Pinchot. But if that person denied authorship of the poem I'm certain that Claire MacMurray must have meant the *first*. ;)
There's also the very likely possibility that the quoted example wasn't the first time she'd been credited with the poem and that she'd denied it at an earlier date. These things never die, even in the face of debunking, and especially not in the face of death of the principals. -- Tom Duff. It puts something of a damper on the discussion of jello howitzers.
I first came across it some time in the 1960's in an anthology of comic verse. A search turns up http://www.coldspur.com/reviews/hogamous-higamous/ which tentatively ascribes it to William James (brother of Henry James): apparently one of his academic investigations during the 1880's involved inhaling laughing gas and other psychotropic agents, while recording the resulting philosophical insights for critical evaluation when sober. It is also suggested that Bertrand Russell may have had a hand in its dissemination to posterity. One way and another, quite a respectable pedigree! WFL On 4/1/16, Dan Asimov <dasimov@earthlink.net> wrote:
Btw, I was certain that this is a takeoff on a rhyme *written by Aldous Huxley*, having long ago read some non-fiction by or about him that stated he made this up as a schoolboy. So googled to confirm this.
But the Internet mainly thinks the origin of this is undetermined (just google on hogamous, higamous). After probably close to 10 minutes of googling I found *one* hit where someone said it was by Aldous Huxley — the others didn't even mention him as a candidate.
And that hit was just a comment by someone in a discussion group, no citation.
Do you, Fred, or anyone else, know the origin of
Hogamous, higamous, Man is polygamous. Higamous, hogamous, Woman monogamous.
?
—Dan
On Apr 1, 2016, at 8:23 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hogamous, higamous, Plouffe is polygammous. Higamous, hogamous, Gosper monogammous.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Many web pages I came upon mentioned the William James theory, but said that had been disproved or was extremely unlikely, for various reasons. I still bet on Aldous Huxley! –Dan
On Apr 1, 2016, at 1:29 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I first came across it some time in the 1960's in an anthology of comic verse.
A search turns up http://www.coldspur.com/reviews/hogamous-higamous/ which tentatively ascribes it to William James (brother of Henry James): apparently one of his academic investigations during the 1880's involved inhaling laughing gas and other psychotropic agents, while recording the resulting philosophical insights for critical evaluation when sober.
It is also suggested that Bertrand Russell may have had a hand in its dissemination to posterity. One way and another, quite a respectable pedigree!
WFL
On 4/1/16, Dan Asimov <dasimov@earthlink.net> wrote:
Btw, I was certain that this is a takeoff on a rhyme *written by Aldous Huxley*, having long ago read some non-fiction by or about him that stated he made this up as a schoolboy. So googled to confirm this.
But the Internet mainly thinks the origin of this is undetermined (just google on hogamous, higamous). After probably close to 10 minutes of googling I found *one* hit where someone said it was by Aldous Huxley — the others didn't even mention him as a candidate.
And that hit was just a comment by someone in a discussion group, no citation.
Do you, Fred, or anyone else, know the origin of
Hogamous, higamous, Man is polygamous. Higamous, hogamous, Woman monogamous.
?
—Dan
On Apr 1, 2016, at 8:23 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hogamous, higamous, Plouffe is polygammous. Higamous, hogamous, Gosper monogammous.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (7)
-
Dan Asimov -
Dan Asimov -
Fred Lunnon -
Hans Havermann -
James Propp -
Simon Plouffe -
Tom Duff