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Jacobi Triple Product

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The Jacobi triple product is the beautiful identity

product_(n=1)^infty(1-x^(2n))(1+x^(2n-1)z^2)(1+(x^(2n-1))/(z^2))=sum_(m=-infty)^inftyx^(m^2)z^(2m).
(1)

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

Q_1Q_2Q_3=1,
(2)

which is one of the two Jacobi identities. In q-series notation, the Jacobi triple product identity is written

(q,-xq,-1/x;q)_infty=sum_(k=-infty)^inftyx^kq^(k(k+1)/2)
(3)

for 0<|q|<1 and x!=0 (Gasper and Rahman 1990, p. 12; Leininger and Milne 1999). Another form of the identity is

sum_(n=-infty)^infty(-1)^na^nq^(n(n-1)/2)=product_(n=1)^infty(1-aq^(n-1))(1-a^(-1)q^n)(1-q^n)
(4)

(Hirschhorn 1999).

Dividing (4) by 1-a and letting a->1 gives the limiting case

(q,q)_infty^3=sum_(n=0)^(infty)(-1)^n(2n+1)q^(n(n+1)/2)
(5)
=1/2sum_(n=-infty)^(infty)(-1)^n(2n+1)q^(n(n+1)/2)
(6)

(Jacobi 1829; Hardy and Wright 1979; Hardy 1999, p. 87; Hirschhorn 1999; Leininger and Milne 1999).

For the special case of z=1, (◇) becomes

theta_3(x)=G(1)
(7)
=product_(n=1)^(infty)(1+x^(2n-1))^2(1-x^(2n))
(8)
=sum_(m=-infty)^(infty)x^(m^2)
(9)
=1+2sum_(m=1)^(infty)x^(m^2),
(10)

where theta_3(x) is a Jacobi elliptic function. In terms of the two-variable Ramanujan theta function f(a,b), the Jacobi triple product is equivalent to

f(a,b)=(-a;ab)_infty(-b;ab)_infty(ab;ab)_infty
(11)

(Berndt et al. 2000).

One method of proof for the Jacobi identity proceeds by defining the function

F(z)=product_(n=1)^(infty)(1+x^(2n-1)z^2)(1+(x^(2n-1))/(z^2))
(12)
=(1+xz^2)(1+x/(z^2))(1+x^3z^2)(1+(x^3)/(z^2))(1+x^5z^2)(1+(x^5)/(z^2))....
(13)

Then

F(xz)=(1+x^3z^2)(1+1/(xz^2))(1+x^5z^2)(1+x/(z^2))×(1+x^7z^2)(1+(x^3)/(z^2))....
(14)

Taking (14) ÷ (13),

(F(xz))/(F(z))=(1+1/(xz^2))(1/(1+xz^2))
(15)
=(xz^2+1)/(xz^2)1/(1+xz^2)=1/(xz^2),
(16)

which yields the fundamental relation

xz^2F(xz)=F(z).
(17)

Now define

G(z)=F(z)product_(n=1)^infty(1-x^(2n))
(18)
G(xz)=F(xz)product_(n=1)^infty(1-x^(2n)).
(19)

Using (17), (19) becomes

G(xz)=(F(z))/(xz^2)product_(n=1)^(infty)(1-x^(2n))
(20)
=(G(z))/(xz^2),
(21)

so

G(z)=xz^2G(xz).
(22)

Expand G in a Laurent series. Since G is an even function, the Laurent series contains only even terms.

G(z)=sum_(m=-infty)^inftya_mz^(2m).
(23)

Equation (22) then requires that

sum_(m=-infty)^(infty)a_mz^(2m)=xz^2sum_(m=-infty)^(infty)a_m(xz)^(2m)
(24)
=sum_(m=-infty)^(infty)a_mx^(2m+1)z^(2m+2).
(25)

This can be re-indexed with m^'=m-1 on the left side of (25)

sum_(m=-infty)^inftya_mz^(2m)=sum_(m=-infty)^inftya_mx^(2m-1)z^(2m),
(26)

which provides a recurrence relation

a_m=a_(m-1)x^(2m-1),
(27)

so

a_1=a_0x
(28)
a_2=a_1x^3=a_0x^(3+1)=a_0x^4=a_0x^(2^2)
(29)
a_3=a_2x^5=a_0x^(5+4)=a_0x^9=a_0x^(3^2).
(30)

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))/2-m=m^2.
(31)

Therefore,

a_m=a_0x^(m^2).
(32)

This means that

G(z)=a_0sum_(m=-infty)^inftyx^(m^2)z^(2m).
(33)

The coefficient a_0 must be determined by going back to (◇) and (◇) and letting z=1. Then

F(1)=product_(n=1)^(infty)(1+x^(2n-1))(1+x^(2n-1))
(34)
=product_(n=1)^(infty)(1+x^(2n-1))^2
(35)
G(1)=F(1)product_(n=1)^(infty)(1-x^(2n))
(36)
=product_(n=1)^(infty)(1+x^(2n-1))^2product_(n=1)^(infty)(1-x^(2n))
(37)
=product_(n=1)^(infty)(1+x^(2n-1))^2(1-x^(2n)),
(38)

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,
(39)

so we have the Jacobi triple product,

G(z)=product_(n=1)^(infty)(1-x^(2n))(1+x^(2n-1)z^2)(1+(x^(2n-1))/(z^2))
(40)
=sum_(m=-infty)^(infty)x^(m^2)z^(2m).
(41)

See also

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

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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 (q)_infty^(n^2+n) and Basic Hypergeometric Series in U(n)." 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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Jacobi Triple Product

Cite this as:

Weisstein, Eric W. "Jacobi Triple Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiTripleProduct.html

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