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Jacobi Triple Product

The Jacobi triple product is the beautiful identity

\prod_{n=1}^\infty (1-x^{2n})(1+x^{2n-1}z^2)\left({1+{x^{2n-1}\over z^2}}\right)= \sum_{m=-\infty }^\infty x^{m^2}z^{2m}.
\end{displaymath} (1)

In terms of the Q-Function, (1) is written
\end{displaymath} (2)

which is one of the two Jacobi Identities. For the special case of $z = 1$, (1) becomes
$\displaystyle \varphi(x)$ $\textstyle \equiv$ $\displaystyle G(1)=\prod_{n=1}^\infty (1+x^{2n-1})^2(1-x^{2n})$  
  $\textstyle =$ $\displaystyle \sum_{m=-\infty}^\infty x^{m^2} = 1+2\sum_{m=1}^\infty x^{m^2},$ (3)

where $\varphi(x)$ is the one-variable Ramanujan Theta Function.

To prove the identity, define the function

$\displaystyle F(z)$ $\textstyle \equiv$ $\displaystyle \prod_{n=1}^\infty (1+x^{2n-1}z^2)\left({1 + {x^{2n-1}\over z^2}}\right)$  
  $\textstyle =$ $\displaystyle (1+xz^2)\left({1 + {x\over z^2}}\right)(1+x^3z^2)\left({1 + {x^3\over z^2}}\right)(1+x^5z^2)\left({1 + {x^5\over z^2}}\right)\cdots.$ (4)


F(xz) = (1+x^3z^2)\left({1 + {1\over xz^2}}\right)(1+x^5z^2)...
...z^2}}\right)(1+x^7z^2)\left({1 + {x^3\over z^2}}\right)\cdots.
\end{displaymath} (5)

Taking (5) $\div$ (4),
$\displaystyle {F(xz)\over F(z)}$ $\textstyle =$ $\displaystyle \left({1 + {1\over xz^2}}\right)\left({1\over 1+xz^2}\right)$  
  $\textstyle =$ $\displaystyle {xz^2+1\over xz^2} {1\over 1+xz^2} = {1\over xz^2},$ (6)

which yields the fundamental relation
xz^2F(xz) = F(z).
\end{displaymath} (7)

Now define
G(z) \equiv F(z)\prod_{n=1}^\infty (1-x^{2n})
\end{displaymath} (8)

G(xz) = F(xz) \prod_{n=1}^\infty (1-x^{2n}).
\end{displaymath} (9)

Using (7), (9) becomes
G(xz) = {F(z)\over xz^2} \prod_{n=1}^\infty (1-x^{2n}) = {G(z)\over xz^2},
\end{displaymath} (10)

G(z) = xz^2G(xz).
\end{displaymath} (11)

Expand $G$ in a Laurent Series. Since $G$ is an Even Function, the Laurent Series contains only even terms.
G(z) =\sum_{m=-\infty}^\infty a_mz^{2m}.
\end{displaymath} (12)

Equation (11) then requires that
$\displaystyle \sum_{m=-\infty}^\infty a_mz^{2m}$ $\textstyle =$ $\displaystyle xz^2\sum_{m=-\infty }^\infty a_m(xz)^{2m}$  
  $\textstyle =$ $\displaystyle \sum_{m=-\infty}^\infty a_mx^{2m+1}z^{2m+2}.$ (13)

This can be re-indexed with $m'\equiv m-1$ on the left side of (13)
\sum_{m=-\infty}^\infty a_mz^{2m} = \sum_{m=-\infty}^\infty a_mx^{2m-1}z^{2m},
\end{displaymath} (14)

which provides a Recurrence Relation
a_m = a_{m-1}x^{2m-1},
\end{displaymath} (15)

$\displaystyle a_1$ $\textstyle =$ $\displaystyle a_0x$ (16)
$\displaystyle a_2$ $\textstyle =$ $\displaystyle a_1x^3 = a_0x^{3+1} = a_0x^4 =a_0x^{2^2}$ (17)
$\displaystyle a_3$ $\textstyle =$ $\displaystyle a_2x^5 = a_0x^{5+4} = a_0x^9 = a_0x^{3^2}.$ (18)

The exponent grows greater by $(2m-1)$ for each increase in $m$ of 1. It is given by
\sum_{n=1}^m (2m-1) = 2 {m(m+1)\over 2}-m = m^2.
\end{displaymath} (19)

a_m = a_0x^{m^2}.
\end{displaymath} (20)

This means that
G(z) = a_0\sum_{m=-\infty }^\infty x^{m^2}z^{2m}.
\end{displaymath} (21)

The Coefficient $a_0$ must be determined by going back to (4) and (8) and letting $z = 1$. Then
$\displaystyle F(1)$ $\textstyle =$ $\displaystyle \prod_{n=1}^\infty (1+x^{2n-1})(1+x^{2n-1})$  
  $\textstyle =$ $\displaystyle \prod_{n=1}^\infty (1+x^{2n-1})^2$ (22)
$\displaystyle G(1)$ $\textstyle =$ $\displaystyle F(1) \prod_{n=1}^\infty (1-x^{2n})$  
  $\textstyle =$ $\displaystyle \prod_{n=1}^\infty (1+x^{2n-1})^2\prod_{n=1}^\infty (1-x^{2n})$  
  $\textstyle =$ $\displaystyle \prod_{n=1}^\infty (1+x^{2n-1})^2(1-x^{2n}),$ (23)

since multiplication is Associative. It is clear from this expression that the $a_0$ term must be 1, because all other terms will contain higher Powers of $x$. Therefore,
a_0 = 1,
\end{displaymath} (24)

so we have the Jacobi triple product,
$\displaystyle G(z)$ $\textstyle =$ $\displaystyle \prod_{n=1}^\infty (1-x^{2n})(1+x^{2n-1}z^2)\left({1 + { x^{2n-1}\over z^2}}\right)$  
  $\textstyle =$ $\displaystyle \sum_{m=-\infty }^\infty x^{m^2}z^{2m}.$ (25)

See also Euler Identity, Jacobi Identities, Q-Function, Quintuple Product Identity, Ramanujan Psi Sum, Ramanujan Theta Functions, Schröter's Formula, Theta Function


Andrews, G. E. $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 63-64, 1986.

Borwein, J. M. and Borwein, P. B. ``Jacobi's Triple Product and Some Number Theoretic Applications.'' Ch. 3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 62-101, 1987.

Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum Borntraeger, p. 90, 1829.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 470, 1990.

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© 1996-9 Eric W. Weisstein