January 2013 - mathfun@mailman.xmission.com

[math-fun] product_{0<k<n, (k,n)=1} Gamma(z+k/n) ? [Was: product_{(k,n)=1} Γ(z+k/n) ?]
by Bill Gosper 19 Jan '13

19 Jan '13

wds> Gosper has an annoying habit of using non-ASCII characters I cannot read, which come out as "?" on my screen. Or maybe they really are "?". Who knows? hmm, that might explain why I'm not getting any answers. Off list I've been merrily exchanging emails with Chinese and all sorts of exotica from the Special Characters menu. But this time the culprit is not math-fun--I can see the Uppercase Gamma in my Subject summary window, where there's a ? in Warren's reply. Who else is having this problem? Was anybody besides Warren pwned by my BLACK RIGHT-POINTING SMALL TRIANGLE (▸), (which GMail has suddenly stopped inserting on double-clicking the font table)? Maybe Warren is alone on a network made of juice cans and string? Anyway, he's right. I should've stipulated 0<k<n under the product. But he might have guessed my intent from the n=12 example, which is plain ASCII, except for the pi. Haven't we been using π quite a while now? It's just option p on the Mac keyboard. --rwg On Fri, Jan 18, 2013 at 3:39 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > Someone once sent me (us?) a paper deriving (presumably by > inclusion-exclusion) > this gamma product formula. Does anyone remember where it is? Presumably > for n=12 it gives something like > > (z - 11/12)!*(z - 7/12)!*(z - 5/12)!*(z - 1/12)!== > 4*(2*z)!*(12*z)!*π^2/(2^(12*z)*3^(6*z)*(4*z)!*(6*z)!) > > It doesn't seem to be on http://functions.wolfram.com/ > --rwg >

6 5

Re: [math-fun] readability of math-fun posts
by Axel Vogt 19 Jan '13

19 Jan '13

> Message: 4 > Date: Sat, 19 Jan 2013 14:42:51 -0500 > From: Warren Smith<warren.wds(a)gmail.com> > To:math-fun@mailman.xmission.com > Subject: [math-fun] readability of math-fun posts > Message-ID: > <CAAJP7Y1+=TqpHOuBbmdTdSVNoHFj83ZJNN882FeSRE6DifcXfw(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > --If you stick to plain ASCII character set, then you are safe. If > you use anything else, > then some people will have trouble reading it. Sure, we may be able > to compensate > if we try hard enough, but I think a post for a general, > cross-platform audience, ought to > be designed so Ignorant Joe Schmoe can read it. Another thing some > posters enjoy doing is making a fairly long message which is all on > one single line, say 1000 characters long. > Again, some reader tools insert line breaks to make that readable. > Others do not. I am still too dumb to answer correctly (using Thunderbird as mailer) into a thread :-( But it may be, that for example Bill Gosper (using Google Mail) may have some setting to provide html mail and/or encoding is not the prefered one (ISO-8859-1).

1 0

[math-fun] Gamma product request, inclusion-exclusion & other residue tricks, and readability
by Warren Smith 19 Jan '13

19 Jan '13

So is what Gosper now wants, this?: PRODUCT(over n & k with n>0, 0<k<n, gcd(k,n)=1)OF (z-k/n) or is it this?: PRODUCT(over k with 0<k<n, gcd(k,n)=1)OF (z-k/n)! Anyway, about inclusion exclusion and other games with residues mod n: A. I recently rediscovered, which was known to Gauss, the facts that the residues mod n which are relatively prime to n, (1) form a multiplicative abelian group, (2) which is cyclic if and only if n is odd-prime-power, 2 times odd-prime-power, or n=2 or n=4, (3) which is "bicyclic" if n>4 is a power of 2, that is, all such residues are powers of 5 times +1 or -1, and the powers of 5 times +1 constitute a cyclic subgroup of half the full size, where full size of course is EulerTotient(n). These facts often allow interesting results to be got for these "nice" n. B. Assume n is squarefree for simplicity. Let "weight(x)" be the number of primes in x's factorization. Use the fact that SUM(0<=k<=n) Binomial(n,k) * (-1)^k = (1-1)^n = 0. Conclude SUM(divisors d of n) (-1)^weight(d) = 0. Also SUM(divisors d of n with d>1) (-1)^weight(d) = -1. The application of this is SUM(k with 0<k<n and gcd(k,n)=1)OF something = SUM(divisors d of n) (-1)^weight(d) * SUM(k with 0<k<n and d divides k) something C. Products are sums if you use logs. That is, B and C allow you to do inclusion-exclusion where everything involved is a sum over an additive subgroup of residues mod n, i.e. nice. The function (-1)^weight(d) more generally can be replaced by MobiusFunction(d) if we allow n that are not squarefree -- that will work for many uses. > Gosper has an annoying habit of using non-ASCII > characters I cannot read, which come out as "?" > on my screen. Or maybe they really are "?". Who knows? > > hmm, that might explain why I'm not getting any answers. Off list I've > been merrily exchanging > emails with Chinese and all sorts of exotica from the Special Characters > menu. But this > time the culprit is not math-fun--I can see the Uppercase Gamma in my > Subject summary > window, where there's a ? in Warren's reply. > > Who else is having this problem? Was anybody besides Warren pwned by > my BLACK RIGHT-POINTING SMALL TRIANGLE (?), (which GMail has > suddenly stopped inserting on double-clicking the font table)? > > Maybe Warren is alone on a network made of juice cans and string? > > Anyway, he's right. I should've stipulated 0<k<n under the product. But > he might have > guessed my intent from the n=12 example, which is plain ASCII, except for > the pi. > Haven't we been using ? quite a while now? It's just option p on the Mac > keyboard. > --rwg --Your ? here came out as ? on my reader. I have a mac. When I type "option-p" into this here window, I do not get pi. I get p. I just did it. Gosper's posts often are hard for me to read, they often have had wrong line-breaks and unreadable characters combined. The solution is to post using only ASCII characters and manually placing all your line breaks exactly where they should be, using fairly short lines only --not depending on tools to insert them for you (since they may place them wrong). If you do that, everybody can read it. If you do not, some people will have trouble reading it. Your choice.

1 0

Re: [math-fun] playing KITP videos with VLC
by Henry Baker 19 Jan '13

19 Jan '13

OK, I finally got these very-low-frame-rate videos to play properly with the latest VLC player. Many of the videos that you can download from the UCSB KITP site are encoded at .1 fps (yes, that's 1/10 of a frame per second, or a new frame every 10 secs.). The VLC player _can_ handle these videos, but you have to change the "File caching" parameter to 30000 ms or more -- I set mine to 60000 ms = 60 _seconds_ of file caching. (This parameter doesn't want to go above 60 seconds.) Note that this setting doesn't harm latency, but might require enormous amounts of memory for "standard" frame rate video at 30fps or 60fps. BTW, I did some tests at a number of different frame rates & file caching parameter settings, and determined that if F=# frames per second, and C=file caching parameter in seconds, then F*C >= 3 seconds seems to work. In this case, F=0.1 and C=30 means that 0.1*30=3.0 >= 3, and it works. (IMHO this is a bug in VLC; the amount of file caching shouldn't be specified in terms of seconds, but in terms of the number of cached frames.) At 06:44 PM 1/13/2013, Robert Munafo wrote: >On 1/13/13, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I'm running VLC 2.0.5 on both XP and Windows 7. Neither one shows the >> video; the audio plays fine. > >Wow, that's a much bigger version number. (-: > >> I did a Google search, and VLC's problem with low frame rate video (< 3fps) >> appears to be well known. >> >> Do you know any magic preferences settings for VLC to make .1fps video's work? > >At http://forum.videolan.org/viewtopic.php?f=2&t=103264 a user >suggests: "Tools => Preferences => Input & Codecs => Default caching >policy => Higher latency => Save => Quit" (and then I guess, restart >VLC and try playing the video) > >Since it involves cacheing, I'll point out that I've done this in my >VLC which might be why mine works. I use the detailed settings: >Prefences => All (little button in lower-left corner) => Input/Codecs >=> Access Modules => File => Cacheing value and Extra network cacheing >value both set to 1000 ms. > >Also, why not try a VLC version closer to my version 1.1.10.1? > > http://download.videolan.org/pub/videolan/vlc/1.1.10/win32/ > >It's only about 1.5 years old, and the great thing about Windows is >that, (unlike Apple), Microsoft actually gives a $#&* about backwards >compatibility, so you'll probably be able to run it for years to come. > >I recommend you keep multiple old versions of VLC around, because >"newer" seldom means "better". > >-- > Robert Munafo -- mrob.com > Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - >mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com

1 0

[math-fun] readability of math-fun posts
by Warren Smith 19 Jan '13

19 Jan '13

--If you stick to plain ASCII character set, then you are safe. If you use anything else, then some people will have trouble reading it. Sure, we may be able to compensate if we try hard enough, but I think a post for a general, cross-platform audience, ought to be designed so Ignorant Joe Schmoe can read it. Another thing some posters enjoy doing is making a fairly long message which is all on one single line, say 1000 characters long. Again, some reader tools insert line breaks to make that readable. Others do not.

1 0

[math-fun] Ambigrams
by Adam P. Goucher 19 Jan '13

19 Jan '13

I made an ambigram in first-order logic equipped with logical and arithmetic operators. See the second section of this post: http://cp4space.wordpress.com/2013/01/17/assorted-stuff/ It's a silly example, but it's still grammatically correct and non-trivial. Sincerely, Adam P. Goucher

1 0

[math-fun] OEIS: conjecture about certain forests and their exponential generating functions
by Joerg Arndt 19 Jan '13

19 Jan '13

I'd like to know whether the following is known (I bet it is, and in a more general context): cf. https://oeis.org/A187761 and https://oeis.org/A179455 Let C(n,x) be the e.g.f of the monotonic-labeled forests on n vertices with rooted trees of height less than n. (A labeled rooted tree is monotonic-labeled if the label of any parent vertex is (strictly) smaller than the label of any offspring vertex.) I conjecture that the C(n,x) are as follows. Let C(0,x) = 1 and for n>=1, C(n, x) = exp(integral(C(n-1,x)) ) For n=1 (C(1,x)=exp(x), constant sequence of ones) and n=2 (C(2,x)=exp(exp(x)-1), Bell numbers) this is obviously true. (note that integration of an e.g.f. essentially shifts it to smaller powers and drops the constant term; for the corresponding sequences: it just drops the first term). So clueless, jj

2 5

[math-fun] product_{(k,n)=1} Γ(z+k/n) ?
by Bill Gosper 19 Jan '13

19 Jan '13

Someone once sent me (us?) a paper deriving (presumably by inclusion-exclusion) this gamma product formula. Does anyone remember where it is? Presumably for n=12 it gives something like (z - 11/12)!*(z - 7/12)!*(z - 5/12)!*(z - 1/12)!== 4*(2*z)!*(12*z)!*π^2/(2^(12*z)*3^(6*z)*(4*z)!*(6*z)!) It doesn't seem to be on http://functions.wolfram.com/ --rwg

2 1

[math-fun] product_{(k,n)=1} ?(z+k/n) ?
by Warren Smith 18 Jan '13

18 Jan '13

Gosper has an annoying habit of using non-ASCII characters I cannot read, which come out as "?" on my screen. Or maybe they really are "?". Who knows? Anyhow, product_{(k,n)=1} (z+k/n) clearly equals infinity and/or undefined if z>0. [I assume (k,n) means gcd(k,n)... I assume k>0 and n>0 intended... you know Bill, there is nothing actually wrong with writing math in a way intended to make it clearly easily readable+unambiguous...] To see this, pair up terms with k/n and n/k. On 1/18/13, math-fun-request(a)mailman.xmission.com <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: Best explanation of the Higgs particle (Wouter Meeussen) > 2. Re: Best explanation of the Higgs particle (Henry Baker) > 3. Re: Best explanation of the Higgs particle > (Meeussen Wouter (bkarnd)) > 4. why string theory can have high-spin particles without > inconsistency (allegedly) (Warren Smith) > 5. Joseph Smith's favorite function (Bill Gosper) > 6. product_{(k,n)=1} ?(z+k/n) ? (Bill Gosper) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Wed, 16 Jan 2013 23:57:03 +0100 > From: "Wouter Meeussen" <wouter.meeussen(a)telenet.be> > To: "math-fun" <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Best explanation of the Higgs particle > Message-ID: <35C36383F9584D8B92EBA539D0527D04@Wou> > Content-Type: text/plain; format=flowed; charset="iso-8859-1"; > reply-type=original > > along similar lines, > > playing with http://xaonon.dyndns.org/hawking/hrcalc.js > I came unto a surprise: > fill in the estimated mass 3.4 10^54 kg of the the known (?) universe into > the formula for a black hole, > and estimate the diameter (or circumference/Pi for you singularophobics out > > there), > what diameter do you get? > > to be (inside) or not to be (inside), that's the question > > Wouter. > > > -----Original Message----- > From: Warren Smith > Sent: Wednesday, January 16, 2013 2:39 AM > To: math-fun > Subject: [math-fun] Best explanation of the Higgs particle > >> Here's the download link for the full-motion video: > http://online.itp.ucsb.edu/download/bblunch/arkanihamed2/snd/Arkanihamed2_B… > > --I listened to this Nima Arkani-Hamed lecture, my computer was able > to handle it > although took an hour to download. > > One thing N.A-H mentioned as a key building block > was that it is impossible for a particle to have spin>2. > > However, in "string theory" just spin a string harder. You can get > whatever spin you want, no matter how large. No? > > So: How are these two claims to be reconciled? > > [I have asked this question of physicists before, by the way. So far, > I have never got any answer back I considered reasonable.] > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > ------------------------------ > > Message: 2 > Date: Wed, 16 Jan 2013 15:42:37 -0800 > From: Henry Baker <hbaker1(a)pipeline.com> > To: Wouter Meeussen <wouter.meeussen(a)telenet.be> > Cc: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Best explanation of the Higgs particle > Message-ID: <E1TvceG-0005Tc-LI(a)elasmtp-banded.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > Interesting. I got 530 billion light years for the radius. Is this > correct? > > This is only ~37x the current estimated size of the universe, so at least > the orders of magnitude are close. Perhaps it is the estimate of the mass > of the universe that is off? > > Perhaps we're inside the black hole, and that microwave background we see is > our own event horizon? > > BTW, Planck's initial insight into the solution to the "ultraviolet > catastrophe" (I would have called it the "purple paradox", but I guess that > phrase wasn't dramatic enough for the popular press) was to posit energy in > quanta. > > Then Rutherford developed the "planetary model" of the atom, but the problem > with this is that a classical electron would almost immediately radiate way > all of its orbital energy. The solution was that electrons could be in > resonant "orbits" which didn't radiate (or more accurately, would radiate in > particular energy jumps; for long-lived atoms, these jumps were far larger > than normally available). > > I suspect that something similar may be happening to black holes at the > smallest sizes -- there may be resonances which allow very small black holes > to survive far longer than the standard Hawking radiation model would > predict. > > At 02:57 PM 1/16/2013, Wouter Meeussen wrote: >>along similar lines, >> >>playing with http://xaonon.dyndns.org/hawking/hrcalc.js >>I came unto a surprise: >>fill in the estimated mass 3.4 10^54 kg of the the known (?) universe into >> the formula for a black hole, >>and estimate the diameter (or circumference/Pi for you singularophobics out >> there), >>what diameter do you get? >> >>to be (inside) or not to be (inside), that's the question >> >>Wouter. > > > > > > ------------------------------ > > Message: 3 > Date: Thu, 17 Jan 2013 09:12:36 +0100 > From: "Meeussen Wouter (bkarnd)" <wouter.meeussen(a)vandemoortele.com> > To: 'math-fun' <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Best explanation of the Higgs particle > Message-ID: <C6CF6AA22326BC4FA796A688AFF8551A17887C0D38(a)MAIL.vdm.grp> > Content-Type: text/plain; charset="us-ascii" > > >> This is only ~37x the current estimated size of the universe > > that's what I got too. > No surprise in hindsight, since we assume the universe's expansion speed to > equal c if we multiply the uniform expansion rate by the estimated universe > size. > > Some things do no grok though: > temperature seen from the inside (3K cosmic background) doen't match that > seen 'from the outside'; > > Wouter. > (cosmology from the hip, tongue in cheek) > > > -----Original Message----- > From: math-fun-bounces(a)mailman.xmission.com > [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Henry Baker > Sent: donderdag 17 januari 2013 0:43 > To: Wouter Meeussen > Cc: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Best explanation of the Higgs particle > > Interesting. I got 530 billion light years for the radius. Is this > correct? > > This is only ~37x the current estimated size of the universe, so at least > the orders of magnitude are close. Perhaps it is the estimate of the mass > of the universe that is off? > > Perhaps we're inside the black hole, and that microwave background we see is > our own event horizon? > > BTW, Planck's initial insight into the solution to the "ultraviolet > catastrophe" (I would have called it the "purple paradox", but I guess that > phrase wasn't dramatic enough for the popular press) was to posit energy in > quanta. > > Then Rutherford developed the "planetary model" of the atom, but the problem > with this is that a classical electron would almost immediately radiate way > all of its orbital energy. The solution was that electrons could be in > resonant "orbits" which didn't radiate (or more accurately, would radiate in > particular energy jumps; for long-lived atoms, these jumps were far larger > than normally available). > > I suspect that something similar may be happening to black holes at the > smallest sizes -- there may be resonances which allow very small black holes > to survive far longer than the standard Hawking radiation model would > predict. > > At 02:57 PM 1/16/2013, Wouter Meeussen wrote: >>along similar lines, >> >>playing with http://xaonon.dyndns.org/hawking/hrcalc.js >>I came unto a surprise: >>fill in the estimated mass 3.4 10^54 kg of the the known (?) universe >>into the formula for a black hole, and estimate the diameter (or >>circumference/Pi for you singularophobics out there), what diameter do you >> get? >> >>to be (inside) or not to be (inside), that's the question >> >>Wouter. > > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > =============================== > This email is confidential and intended solely for the use of the individual > to whom it is addressed. > If you are not the intended recipient, be advised that you have received > this email in error and that any use, dissemination, forwarding, printing, > or copying of this email is strictly prohibited. > You are explicitly requested to notify the sender of this email that the > intended recipient was not reached. > > > > > ------------------------------ > > Message: 4 > Date: Thu, 17 Jan 2013 16:25:38 -0500 > From: Warren Smith <warren.wds(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] why string theory can have high-spin particles > without inconsistency (allegedly) > Message-ID: > <CAAJP7Y0ZCy__+UbPBzz3zsFFjhM2hGxehF3refjvkZQmNXfK2g(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > There are various arguments, some dating back to the 1960s, which show > that quantum field theories cannot allow any fundamental particle to > exist with spin>2. But "string theory" has arbitrarily high spin > "particles." Contradiction? > > I had asked several physicists about this over many years & none > provided a good answer. However, allegedly the answer is this. > > The inconsistency arguments assume (sometimes implicitly) that the > number of particle/field types in the quantum field theory is finite. > String theory escapes by having infinite set. Essentially, each time > there would be an inconsistency, it is canceled out by putting in an > interaction with even-higher spin particles which cancel out the > inconsistency. Due to infiniteness, the chain of such > fudge-corrections need never end. > > Still, I think this represents a very important/frightening way in > which string theory is fundamentally different from quantum field > theories, deserving of note. This all is not mentioned in most (all?) > of the popular books "explaining" string theory to the unwashed. > > > > ------------------------------ > > Message: 5 > Date: Fri, 18 Jan 2013 04:47:06 -0800 > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Joseph Smith's favorite function > Message-ID: > <CAA-4O0GHU3adPJ6L0GjMa3X=WEUQwZ07-w+kZtYmKm40Q505bw(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-7 > > Can Maple do these? Mma 9.0 and Macsyma can't. Not even half the special > cases. > Out[1015]= > PolyGamma[n, 1/8] + (-1)^n PolyGamma[n, 3/8] == > 2^(1 + n) PolyGamma[n, 1/4] - (2 I ?)^(1 + n) > PolyLog[-n, -((1 + I)/Sqrt[2])] > > n>0. > > In[1016]:= Table[%, {n, 0, 4}] > > In[1017]:= FullSimplify[%] > > Out[1017]= {False, > 32 Catalan + 2 Sqrt[2] ?^2 + PolyGamma[1, 3/8] == > PolyGamma[1, 1/8], > 0 == 12 Sqrt[2] ?^3 + PolyGamma[2, 1/8] + PolyGamma[2, 3/8] + > 448 Zeta[3], > 8 (-16 + 11 Sqrt[2]) ?^4 + 16 PolyGamma[3, 1/4] + > PolyGamma[3, 3/8] == PolyGamma[3, 1/8], > 0 == 912 Sqrt[2] ?^5 + PolyGamma[4, 1/8] + PolyGamma[4, 3/8] + > 380928 Zeta[5]} > > Out[1024]= (-1 + 2^(-1 - n)) PolyGamma[n, 1/6] + > ((-1)^n 2^(-1 - n) + 2^(1 + n)) PolyGamma[n, 1/3] == > I^(-1 + n) ?^(1 + n) PolyLog[-n, 1/2 (-1 - I Sqrt[3])] - (-1)^ > n (-1 + 3^(1 + n)) n! Zeta[1 + n] > > n>0. > > In[1025]:= Table[%, {n, 0, 4}] > > In[1026]:= FullSimplify[%] > > Out[1026]= {False, > 4 ?^2 + 3 PolyGamma[1, 1/6] == 15 PolyGamma[1, 1/3], True, > 16 ?^4 + 3 PolyGamma[3, 1/6] == 51 PolyGamma[3, 1/3], True} > > --rwg > > > ------------------------------ > > Message: 6 > Date: Fri, 18 Jan 2013 15:39:44 -0800 > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] product_{(k,n)=1} ?(z+k/n) ? > Message-ID: > <CAA-4O0GimGZRhYOvwVZ015ntba6J+mmVdZDQ8JOz=5Z-vXQmWQ(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-7 > > Someone once sent me (us?) a paper deriving (presumably by > inclusion-exclusion) > this gamma product formula. Does anyone remember where it is? Presumably > for n=12 it gives something like > > (z - 11/12)!*(z - 7/12)!*(z - 5/12)!*(z - 1/12)!== > 4*(2*z)!*(12*z)!*?^2/(2^(12*z)*3^(6*z)*(4*z)!*(6*z)!) > > It doesn't seem to be on http://functions.wolfram.com/ > --rwg > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 119, Issue 25 > ***************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

[math-fun] Joseph Smith's favorite function
by Bill Gosper 18 Jan '13

18 Jan '13

Can Maple do these? Mma 9.0 and Macsyma can't. Not even half the special cases. Out[1015]= PolyGamma[n, 1/8] + (-1)^n PolyGamma[n, 3/8] == 2^(1 + n) PolyGamma[n, 1/4] - (2 I π)^(1 + n) PolyLog[-n, -((1 + I)/Sqrt[2])] n>0. In[1016]:= Table[%, {n, 0, 4}] In[1017]:= FullSimplify[%] Out[1017]= {False, 32 Catalan + 2 Sqrt[2] π^2 + PolyGamma[1, 3/8] == PolyGamma[1, 1/8], 0 == 12 Sqrt[2] π^3 + PolyGamma[2, 1/8] + PolyGamma[2, 3/8] + 448 Zeta[3], 8 (-16 + 11 Sqrt[2]) π^4 + 16 PolyGamma[3, 1/4] + PolyGamma[3, 3/8] == PolyGamma[3, 1/8], 0 == 912 Sqrt[2] π^5 + PolyGamma[4, 1/8] + PolyGamma[4, 3/8] + 380928 Zeta[5]} Out[1024]= (-1 + 2^(-1 - n)) PolyGamma[n, 1/6] + ((-1)^n 2^(-1 - n) + 2^(1 + n)) PolyGamma[n, 1/3] == I^(-1 + n) π^(1 + n) PolyLog[-n, 1/2 (-1 - I Sqrt[3])] - (-1)^ n (-1 + 3^(1 + n)) n! Zeta[1 + n] n>0. In[1025]:= Table[%, {n, 0, 4}] In[1026]:= FullSimplify[%] Out[1026]= {False, 4 π^2 + 3 PolyGamma[1, 1/6] == 15 PolyGamma[1, 1/3], True, 16 π^4 + 3 PolyGamma[3, 1/6] == 51 PolyGamma[3, 1/3], True} --rwg

1 0