“Jacobi Triple Product”的概念、定义、习题解题方法及翻译-数学辞典

单词

Jacobi Triple Product

释义

Jacobi Triple Product

The Jacobi triple product is the beautiful identity

(1)

In terms of the Q-Function, (1) is written

(2)

which is one of the two Jacobi Identities. For the special case of

, (1) becomes


			(3)

where

is the one-variable Ramanujan Theta Function.

To prove the identity, define the function


			(4)

Then

(5)

Taking (5)

(4),


			(6)

which yields the fundamental relation

(7)

Now define

(8)

(9)

Using (7), (9) becomes

(10)

(11)

Expand

in a Laurent Series. Since

is an Even Function, the Laurent Series contains only even terms.

(12)

Equation (11) then requires that


			(13)

This can be re-indexed with

on the left side of (13)

(14)

which provides a Recurrence Relation

(15)

			(16)
			(17)
			(18)

The exponent grows greater by

for each increase in

of 1. It is given by

(19)

Therefore,

(20)

This means that

(21)

The Coefficient

must be determined by going back to (4) and (8) and letting

. Then


			(22)


			(23)

since multiplication is Associative. It is clear from this expression that the

term must be 1, because allother terms will contain higher Powers of

. Therefore,

(24)

so we have the Jacobi triple product,


			(25)

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

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.

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