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q-Product

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Define the nome by

q=e^(-piK^'(k)/K(k))=e^(ipitau),
(1)

where K(k) is the complete elliptic integral of the first kind with modulus k, K^'(k)=K(sqrt(1-k^2)) is the complementary complete elliptic integral of the first kind, and tau is the half-period ratio. Then the Q-products (sometimes written using a lowercase q instead of a capital Q) are defined by

Q_0(q)=product_(n=1)^(infty)(1-q^(2n))
(2)
=(q^2;q^2)_infty
(3)
Q_1(q)=product_(n=1)^(infty)(1+q^(2n))
(4)
=1/2(-1;q^2)_infty
(5)
Q_2(q)=product_(n=1)^(infty)(1+q^(2n-1))
(6)
=q/(q+1)(-q^(-1);q^2)_infty
(7)
Q_3(q)=product_(n=1)^(infty)(1-q^(2n-1))
(8)
=q/(q-1)(q^(-1);q^2)_infty.
(9)

These are written by Zucker (1990) as w=Q_0, x=Q_1, y=Q_2, and z=Q_3.

The Q-products also satisfy the identities

Q_0(q)Q_1(q)=Q_0(q^2)
(10)
Q_0(q)Q_3(q)=Q_0(q^(1/2))
(11)
Q_2(q)Q_3(q)=Q_3(q^2)
(12)
Q_1(q)Q_2(q)=Q_1(q^(1/2)).
(13)

See also

Dedekind Eta Function, q-Pochhammer Symbol, q-Series Identities

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References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 55 and 63-85, 1987.Tannery, J. and Molk, J. Elements de la Théorie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars et fils, 1893-1902.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 469-473 and 488-489, 1990.Zucker, J. "Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums." J. Phys. A: Math. Gen. 23, 117-132, 1990.

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q-Product

Cite this as:

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

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