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Date: 22-5-2019
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Date: 25-5-2019
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Date: 24-3-2019
1109
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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) |
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
These are written by Zucker (1990) as , , , and .
The -products also satisfy the identities
(10) |
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(11) |
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(12) |
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(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.
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