q-Product
المؤلف:
Borwein, J. M. and Borwein, P. B
المصدر:
Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley
الجزء والصفحة:
...
19-8-2019
2092
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
These are written by Zucker (1990) as 
, 
, 
, and 
.
The 
-products also satisfy the identities
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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