The Jacobi triple product is the beautiful identity
![\begin{displaymath}
\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}](j_386.gif) |
(1) |
In terms of the Q-Function, (1) is written
![\begin{displaymath}
Q_1Q_2Q_3=1,
\end{displaymath}](j_255.gif) |
(2) |
which is one of the two Jacobi Identities. For the special case of
, (1) becomes
where
is the one-variable Ramanujan Theta Function.
To prove the identity, define the function
Then
![\begin{displaymath}
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}](j_396.gif) |
(5) |
Taking (5)
(4),
which yields the fundamental relation
![\begin{displaymath}
xz^2F(xz) = F(z).
\end{displaymath}](j_401.gif) |
(7) |
Now define
![\begin{displaymath}
G(z) \equiv F(z)\prod_{n=1}^\infty (1-x^{2n})
\end{displaymath}](j_402.gif) |
(8) |
![\begin{displaymath}
G(xz) = F(xz) \prod_{n=1}^\infty (1-x^{2n}).
\end{displaymath}](j_403.gif) |
(9) |
Using (7), (9) becomes
![\begin{displaymath}
G(xz) = {F(z)\over xz^2} \prod_{n=1}^\infty (1-x^{2n}) = {G(z)\over xz^2},
\end{displaymath}](j_404.gif) |
(10) |
so
![\begin{displaymath}
G(z) = xz^2G(xz).
\end{displaymath}](j_405.gif) |
(11) |
Expand
in a Laurent Series. Since
is an Even Function, the Laurent Series contains only even terms.
![\begin{displaymath}
G(z) =\sum_{m=-\infty}^\infty a_mz^{2m}.
\end{displaymath}](j_407.gif) |
(12) |
Equation (11) then requires that
This can be re-indexed with
on the left side of (13)
![\begin{displaymath}
\sum_{m=-\infty}^\infty a_mz^{2m} = \sum_{m=-\infty}^\infty a_mx^{2m-1}z^{2m},
\end{displaymath}](j_412.gif) |
(14) |
which provides a Recurrence Relation
![\begin{displaymath}
a_m = a_{m-1}x^{2m-1},
\end{displaymath}](j_413.gif) |
(15) |
so
The exponent grows greater by
for each increase in
of 1. It is given by
![\begin{displaymath}
\sum_{n=1}^m (2m-1) = 2 {m(m+1)\over 2}-m = m^2.
\end{displaymath}](j_421.gif) |
(19) |
Therefore,
![\begin{displaymath}
a_m = a_0x^{m^2}.
\end{displaymath}](j_422.gif) |
(20) |
This means that
![\begin{displaymath}
G(z) = a_0\sum_{m=-\infty }^\infty x^{m^2}z^{2m}.
\end{displaymath}](j_423.gif) |
(21) |
The Coefficient
must be determined by going back to (4) and (8) and letting
. Then
since multiplication is Associative. It is clear from this expression that the
term must be 1, because all
other terms will contain higher Powers of
. Therefore,
![\begin{displaymath}
a_0 = 1,
\end{displaymath}](j_433.gif) |
(24) |
so we have the Jacobi triple product,
See also Euler Identity, Jacobi Identities, Q-Function, Quintuple Product Identity,
Ramanujan Psi Sum, Ramanujan Theta Functions, Schröter's Formula, Theta
Function
References
Andrews, G. E.
-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.
© 1996-9 Eric W. Weisstein
1999-05-25