Quintuple Product Identity

Author

Eric W. Weisstein
August 8, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/QuintupleProductIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/QuintupleProductIdentity.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Expansion

(∏_(n=1)^∞)(1-(q^n)) (1-z (q^n)) (1-((q^(n-1))/z)) (1-(z^2) (q^(2 n-1))) (1-((q^(2 n-1))/(z^2)))==(∑_(m=-∞)^∞)(q^((1/2) m (3 m+1))) ((z^(3 m))-(z^(-3 m-1)))

prod=Product[(1-(q^n))(1-z (q^n))(1-(z^(-1))(q^(n-1)))(1-(z^2)(q^(2n-1)))(1-(z^(-2))(q^(2n-1))),{n,∞}]//FullSimplify

TraditionalForm[%]

Sum[((z^(3m))-(z^(-3m-1)))(q^(m(3m+1)/2)),{m,-∞,∞}]//InputForm

Subtract@@@Table[{prod,Sum[((z^(3m))-(z^(-3m-1)))(q^(m(3m+1)/2)),{m,-150,150}]}/.{q->RandomReal[WorkingPrecision->100],z->RandomComplex[WorkingPrecision->100]},{100}]//Chop//Union

(∏_(j=1)^∞)(1-(q^j)) (((q^j)/z)+1) (z (q^(j-1))+1) (((q^(2 j-1))/(z^2))+1) ((z^2) (q^(2 j-1))+1)==(∑_(k=-∞)^∞)(((-1))^k) (q^((1/2) k (3 k-1))) (z^(3 k)) (z (q^k)+1)

prod=Product[(1-(q^j))(1+(z^(-1))(q^j))(1+z (q^(j-1)))(1+(z^(-2))(q^(2j-1)))(1+(z^2)(q^(2j-1))),{j,1,∞}]

TraditionalForm[%]

Sum[(((-1))^k)(q^(k(3k-1)/2))(z^(3k))(1+z (q^k)),{k,-∞,∞}]//InputForm

Table[{prod,Sum[(((-1))^k)(q^(k(3k-1)/2))(z^(3k))(1+z (q^k)),{k,-150,150}]}/.{q->RandomReal[WorkingPrecision->10],z->RandomComplex[WorkingPrecision->10]},{1}]//Chop//Union

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