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?