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| {{dablink|This is not about the [[mock theta function]]s discovered by Ramanujan.}}
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| In [[mathematics]], particularly [[q-analog]] theory, the '''Ramanujan theta function''' generalizes the form of the Jacobi [[theta function]]s, while capturing their general properties. In particular, the [[Jacobi triple product]] takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after [[Srinivasa Ramanujan]].
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| ==Definition==
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| The Ramanujan theta function is defined as
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| :<math>f(a,b) = \sum_{n=-\infty}^\infty
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| a^{n(n+1)/2} \; b^{n(n-1)/2} </math>
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| for |''ab''| < 1. The [[Jacobi triple product]] identity then takes the form | |
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| :<math>f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math>
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| Here, the expression <math>(a;q)_n</math> denotes the [[q-Pochhammer symbol]]. Identities that follow from this include
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| :<math>f(q,q) = \sum_{n=-\infty}^\infty q^{n^2} =
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| {(-q;q^2)_\infty^2 (q^2;q^2)_\infty} </math>
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| and
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| :<math>f(q,q^3) = \sum_{n=0}^\infty q^{n(n+1)/2} =
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| {(q^2;q^2)_\infty}{(-q; q)_\infty} </math>
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| and
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| :<math>f(-q,-q^2) = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)/2} =
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| (q;q)_\infty </math>
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| this last being the [[Euler function]], which is closely related to the [[Dedekind eta function]]. The Jacobi [[theta function]] may be written in terms of the Ramanujan theta function as:
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| :<math>\vartheta(w, q)=f(qw^2,qw^{-2})</math>
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| ==References==
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| * W.N. Bailey, ''Generalized Hypergeometric Series'', (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
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| * George Gasper and Mizan Rahman, ''Basic Hypergeometric Series, 2nd Edition'', (2004), Encyclopedia of Mathematics and Its Applications, '''96''', Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
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| * {{springer|title=Ramanujan function|id=p/r077200}}
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| *{{Mathworld|RamanujanThetaFunctions|Ramanujan Theta Functions}}
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| [[Category:Q-analogs]]
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| [[Category:Elliptic functions]]
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| [[Category:Theta functions]]
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| [[Category:Srinivasa Ramanujan]]
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