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Quintuple Product Identity

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The quintuple product identity, also called the Watson quintuple product identity, states

product_(n=1)^infty(1-q^n)(1-zq^n)(1-z^(-1)q^(n-1))(1-z^2q^(2n-1))(1-z^(-2)q^(2n-1)) =sum_(m=-infty)^infty(z^(3m)-z^(-3m-1))q^(m(3m+1)/2).
(1)

It can also be written

product_(n=1)^infty(1-q^(2n))(1-q^(2n-1)z)(1-q^(2n-1)z^(-1))(1-q^(4n-4)z^2)(1-q^(4n-4)z^(-2)) =sum_(n=-infty)^inftyq^(3n^2-2n)[(z^(3n)+z^(-3n))-(z^(3n-2)+z^(-(3n-2)))]
(2)

or

sum_(k=-infty)^infty(-1)^kq^((3k^2-k)/2)z^(3k)(1+zq^k) =product_(j=1)^infty(1-q^j)(1+z^(-1)q^j)(1+zq^(j-1))(1+z^(-2)q^(2j-1))(1+z^2q^(2j-1)).
(3)

The quintuple product identity can be written in q-series notation as

sum_(k=-infty)^infty(-1)^kq^(k(3k-1)/2)z^(3k)(1+zq^k)=(1,-z,-q/z;q)_infty(qz^2,q/z^2;q^2)_infty,
(4)

where 0<|q|<1 and z!=0 (Gasper and Rahman 1990, p. 134; Leininger and Milne 1999).

Using the notation of the Ramanujan theta function (Berndt 1985, p. 83),

f(B^3q,q^5/B^3)-B^2f(q/B^3,B^3q^5) =f(-q^2)(f(-B^2,-q^2/B^2))/(f(Bq,q/B)).
(5)

See also

Jacobi Triple Product, Ramanujan Theta Functions, Septuple Product Identity

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References

Berndt, B. C. Ramanujan's Notebooks, Part III. New York:Springer-Verlag, 1985.Bhargava, S. "A Simple Proof of the Quintuple Product Identity." J. Indian Math. Soc. 61, 226-228, 1995.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 306-309, 1987.Carlitz, L. and Subbarao, M. V. "A Simple Proof of the Quintuple Product Identity." Proc. Amer. Math. Soc. 32, 42-44, 1972.Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.Leininger, V. E. and Milne, S. C. "Some New Infinite Families of eta-Function Identities." Methods Appl. Anal. 6, 225-248, 1999b.

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Quintuple Product Identity

Cite this as:

Weisstein, Eric W. "Quintuple Product Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuintupleProductIdentity.html

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