q-Product

Define the nome by

(1)

where  is the complete elliptic integral of the first kind with modulus ,  is the complementary complete elliptic integral of the first kind, and  is the half-period ratio. Then the -products (sometimes written using a lowercase instead of a capital ) are defined by

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

These are written by Zucker (1990) as , , , and .

The -products also satisfy the identities

			(10)
			(11)
			(12)
			(13)

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.