Define the nome by
(1)
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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)
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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)
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