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Date: 16-8-2019
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Ramanujan's two-variable theta function is defined by
(1) |
for (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies
(2) |
and
(3) |
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(4) |
(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) |
(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) |
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) |
(OEIS A000122), where is a Jacobi theta function (Berndt 1985, pp. 36-37). is a generalization of , with the two being connected by
(15) |
Special values of include
(16) |
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(17) |
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) |
(OEIS A010054; Berndt 1985, p. 37).
Ramanujan's -function is defined by
(24) |
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(25) |
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(26) |
(OEIS A000700; Berndt 1985, p. 37).
A different function is sometimes defined as
(27) |
where is again a Jacobi theta function, which has special value
(28) |
REFERENCES:
Berndt, B. C. Ramanujan's Notebooks, Part III. New York: Springer-Verlag, 1985.
Berndt, B. C.; Huang, S.-S.; Sohn, J.; and Son, S. H. "Some Theorems on the Rogers-Ramanujan Continued Fraction in Ramanujan's Lost Notebook." Trans. Amer. Math. Soc. 352, 2157-2177, 2000.
Mc Laughlin, J.; Sills, A. V.; and Zimmer, P. "Dynamic Survey DS15: Rogers-Ramanujan-Slater Type Identities." Electronic J. Combinatorics, DS15, 1-59, May 31, 2008. http://www.combinatorics.org/Surveys/ds15.pdf.
Sloane, N. J. A. Sequences A000122, A000700/M0217, A010054, and A010815 in "The On-Line Encyclopedia of Integer Sequences."
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