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MathGroup Archive 2006

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Re: infinite product

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65309] Re: infinite product
  • From: Ronald Bruck <bruck at math.usc.edu>
  • Date: Sat, 25 Mar 2006 05:17:45 -0500 (EST)
  • References: <6334718.1143190534402.JavaMail.jakarta@nitrogen.mathforum.org> <240320060230425202%bruck@math.usc.edu> <20060324081024.411$gC@newsreader.com>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <20060324081024.411$gC at newsreader.com>, David W. Cantrell
<DWCantrell at sigmaxi.org> wrote:

> Ronald Bruck <bruck at math.usc.edu> wrote:
> > In article
> > <6334718.1143190534402.JavaMail.jakarta at nitrogen.mathforum.org>, eugene
> > <jane1806 at mail.ru> wrote:
> >
> > > Could you please help me to calculate the following product
> > > \prod_{n=2}^{\infty} (n^2-1)/(n^2+1).
> > >
> > > In the case \prod (n^3-1)/(n^3+1) we can easily cancel out the
> > > multipliers in the numerator and denomonator and it can easily be
> > > proved that the values in this case is 2/3. But i have no ideas to deal
> > > with our case with squares.
> >
> > I wouldn't expect this to be anything simple, but I plugged it into
> > Mathematica anyway.  The result may be very interesting to fans of
> > Mathematica:
> >
> > {(E^(LogGamma[2 - I] + LogGamma[2 + I])*Pi*Csch[Pi]*   Gamma[1 +
> > System`SeriesDump`k]^2)/
> > (20*Internal`ErdelyiBernoulliB[System`SeriesDump`k,     3 - I, 2 -
> > I]*Internal`ErdelyiBernoulliB[    System`SeriesDump`k, 3 + I, 2 + I]*
> > Gamma[(-2 - I) + System`SeriesDump`k]*   Gamma[(-2 + I) +
> > System`SeriesDump`k]),  (E^((-2*I)*System`SeriesDump`k*Pi + LogGamma[2
> > - I] +      LogGamma[2 + I])*Gamma[-2 - I]*Gamma[-2 + I]*   Gamma[1 +
> > System`SeriesDump`k]^2)/
> > (2*Internal`ErdelyiBernoulliB[System`SeriesDump`k,     3 - I, 2 -
> > I]*Internal`ErdelyiBernoulliB[    System`SeriesDump`k, 3 + I, 2 + I]*
> > Gamma[(-2 - I) + System`SeriesDump`k]*   Gamma[(-2 + I) +
> > System`SeriesDump`k]),  (E^(LogGamma[2 - I] + LogGamma[2 +
> > I])*Pi*Csch[Pi]*   Gamma[1 + System`SeriesDump`k]^2)/
> > (20*Internal`ErdelyiBernoulliB[System`SeriesDump`k,     3 - I, 2 -
> > I]*Internal`ErdelyiBernoulliB[    System`SeriesDump`k, 3 + I, 2 + I]*
> > Gamma[(-2 - I) + System`SeriesDump`k]*   Gamma[(-2 + I) +
> > System`SeriesDump`k])}
> >
> > Question is, what are "System`SeriesDump`k" and "ErdelyiBernoulliB"?
> > The former is clearly some sort of abort, and the latter an internal
> > routine.
> >
> > The numerical value is approximately 0.272029.
> 
> I've gotten garbage like that from Mathematica before too. But I don't in
> this case, using version 5.1:
> 
> In[1]:= FullSimplify[Product[(n^2 - 1)/(n^2 + 1), {n, 2, Infinity}]]
> 
> Out[1]= Pi*Csch[Pi]
> 
> which agrees with the answer derived earlier by Boudewijn. I wonder what
> version of Mathematica you're using.

OK, I've now tried Version 5.2 on three machines:  two Mac OS X 10.4.5,
and one Windows XP.  All give the same result.

So 5.2 is a slight "downgrade".  I'll add a followup to
comp.soft-sys.math.mathematica.

-- 
Ron Bruck


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