Jacobi Triple Product
المؤلف:
Andrews, G. E
المصدر:
q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc.
الجزء والصفحة:
...
29-9-2019
3818
Jacobi Triple Product
The Jacobi triple product is the beautiful identity
 |
(1)
|
In terms of the Q-functions, (1) is written
 |
(2)
|
which is one of the two Jacobi identities. In q-series notation, the Jacobi triple product identity is written
 |
(3)
|
for 
and 
(Gasper and Rahman 1990, p. 12; Leininger and Milne 1999). Another form of the identity is
 |
(4)
|
(Hirschhorn 1999).
Dividing (4) by 
and letting 
gives the limiting case
(Jacobi 1829; Hardy and Wright 1979; Hardy 1999, p. 87; Hirschhorn 1999; Leininger and Milne 1999).
For the special case of 
, (◇) becomes
where 
is a Jacobi elliptic function. In terms of the two-variable Ramanujan theta function 
, the Jacobi triple product is equivalent to
 |
(11)
|
(Berndt et al. 2000).
One method of proof for the Jacobi identity proceeds by defining the function
Then
Taking (14) 
(13),
which yields the fundamental relation
 |
(17)
|
Now define
 |
(18)
|
 |
(19)
|
Using (17), (19) becomes
so
 |
(22)
|
Expand 
in a Laurent series. Since 
is an even function, the Laurent series contains only even terms.
 |
(23)
|
Equation (22) then requires that
This can be re-indexed with 
on the left side of (25)
 |
(26)
|
which provides a recurrence relation
 |
(27)
|
so
The exponent grows greater by 
for each increase in 
of 1. It is given by
 |
(31)
|
Therefore,
 |
(32)
|
This means that
 |
(33)
|
The coefficient 
must be determined by going back to (◇) and (◇) 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,
 |
(39)
|
so we have the Jacobi triple product,
REFERENCES:
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.
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, p. 222, 2007.
Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan's Lost Notebook." Trans. Amer. Math. Soc. 352, 2157-2177, 2000.
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.
Foata, D. and Han, G.-N. "The Triple, Quintuple and Septuple Product Identities Revisited." In The Andrews Festschrift (Maratea, 1998): Papers from the Seminar in Honor of George Andrews on the Occasion of His 60th Birthday Held in Maratea, August 31-September 6, 1998. Sém. Lothar. Combin. 42, Art. B42o, 1-12, 1999 (electronic).
Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.
Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.
Hirschhorn, M. D. "Another Short Proof of Ramanujan's Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580-583, 1999.
Jacobi, C. G. J. Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, Germany: Regiomonti, Sumtibus fratrum Borntraeger, p. 90, 1829.
Leininger, V. E. and Milne, S. C. "Expansions for 
and Basic Hypergeometric Series in 
." Discr. Math. 204, 281-317, 1999.
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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