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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)

for |x|<1 and z!=0. 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)

for |q|<1 and a!=0 (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, (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 theta 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).

For 0<|x|<1, one method of proof 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)

Dividing the expression for F(xz) by that for F(z) where the denominator is nonzero gives

(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, by continuity, the fundamental relation throughout z!=0,

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 the functional relation for F, this 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)

The infinite product defining G converges uniformly on each compact annulus about the origin, so G has a Laurent series. Since G is an even function, this Laurent series contains only even powers of z. Its coefficients may depend on x.

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

The functional relation for G 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)

Reindexing the right side by replacing its summation index with m-1 gives

sum_(m=-infty)^inftya_mz^(2m)=sum_(m=-infty)^inftya_(m-1)x^(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)

For positive integers m, the exponent increases by 2m-1 at each step and is given by

sum_(n=1)^m(2n-1)=2(m(m+1))/2-m=m^2.
(31)

Therefore,

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

The same formula holds for negative integers m by applying the recurrence relation backwards. This means that

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

Setting z=1 gives

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)

but these evaluations alone do not determine a_0, which is a Laurent series coefficient in z.

To determine a_0, put q=x^2 and consider the finite products

F_N(z)=product_(n=1)^N(1+x^(2n-1)z^2)(1+(x^(2n-1))/(z^2)).
(39)

The finite q-binomial theorem gives

[z^0]F_N(z)=((q;q)_(2N))/((q;q)_N^2),
(40)

where [z^0] is coefficient notation for the constant term and (q;q)_N is a q-Pochhammer symbol (National Institute of Standards and Technology, formulas 17.2.27 and 17.2.35). By the uniform convergence noted above, the constant terms of the finite products converge to those of the infinite product. Consequently,

a_0=lim_(N->infty)(q;q)_N[z^0]F_N(z)=lim_(N->infty)((q;q)_(2N))/((q;q)_N)=1,
(41)

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))
(42)
=sum_(m=-infty)^(infty)x^(m^2)z^(2m).
(43)

For x=0, both sides of the identity equal 1 directly.

See also

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

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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.National Institute of Standards and Technology. "Calculus." §17.2 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/17.2.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 Resource. https://mathworld.wolfram.com/JacobiTripleProduct.html

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