The Jacobi triple product is the beautiful identity

(1) |

(2) |

(3) |

where is the one-variable Ramanujan Theta Function.

To prove the identity, define the function

(4) |

Then

(5) |

(6) |

which yields the fundamental relation

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

This can be re-indexed with on the left side of (13)

(14) |

(15) |

(16) | |||

(17) | |||

(18) |

The exponent grows greater by for each increase in of 1. It is given by

(19) |

(20) |

(21) |

(22) | |||

(23) |

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,

(24) |

(25) |

**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

1999-05-25