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MathGroup Archive 2012

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possible an error in Mathematica or a book proof

  • To: mathgroup at smc.vnet.net
  • Subject: [mg125359] possible an error in Mathematica or a book proof
  • From: Roger Bagula <roger.bagula at gmail.com>
  • Date: Fri, 9 Mar 2012 06:08:56 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

Andrews gives a proof on page 165 that:
George E. Andrews, Number Theory, Dover Publications, N.Y., 1971,
164-165.
Product[1 + x^i, {i, 1, Infinity}]==Product[1/(1 - x^(2*i-1)), {i, 1,
Infinity}]
And this actually checks more generally for rational numbers "a":
Product[1 + x^a*i, {i, 1, Infinity}]==Product[1/(1 - x^(a*2*i-a)), {i,
1, Infinity}]
But if instead a substitution of i->2*i+1:
2(2*i+1)-1=4*i+1
Thus:
Product[1 + x^(2*i + 1), {i, 1, Infinity}] ==Product[1/(1 - x^(4*i +
1)), {i, 1, Infinity}]
But it doesn't in Mathematica!

p[x_] = Product[1 + x^(2*i + 1), {i, 1, Infinity}]
q[x_] = Product[1/(1 - x^(4*i + 1)), {i, 1, Infinity}]
a = Table[SeriesCoefficient[
   Series[p[x], {x, 0, 100}], n], {n, 0, 100}]
ListLinePlot[a]
b = Table[SeriesCoefficient[
   Series[q[x], {x, 0, 100}], n], {n, 0, 100}]
ListLinePlot[b]
a - b

It took a while for it to dawn on me that
it might be a fault in Mathematica
or in the proof.
Since the proof seems to work for rational substitutions
that leaves Mathematica?



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