Ramanujan's two-variable theta function is defined by
(1)
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for (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies
(2)
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and
(3)
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(4)
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(Berndt 1985, pp. 34-35; Berndt et al. 2000), where is a q-Pochhammer symbol, i.e., a q-series.
A one-argument form of is also defined by
(5)
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(6)
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(7)
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(OEIS A010815; Berndt 1985, pp. 36-37; Berndt et al. 2000), where is a q-Pochhammer symbol. The identities above are equivalent to the pentagonal number theorem.
The function also satisfies
(8)
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(9)
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Ramanujan's -function is defined by
(10)
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(11)
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(12)
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(13)
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(14)
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(OEIS A000122), where is a Jacobi theta function (Berndt 1985, pp. 36-37). is a generalization of , with the two being connected by
(15)
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Special values of include
(16)
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(17)
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where is a gamma function.
Ramanujan's -function is defined by
(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(OEIS A010054; Berndt 1985, p. 37).
Ramanujan's -function is defined by
(24)
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(25)
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(26)
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(OEIS A000700; Berndt 1985, p. 37).
A different function is sometimes defined as
(27)
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where is again a Jacobi theta function, which has special value
(28)
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