# Ramanujan theta function

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Ramanujan theta function

In mathematics, the Ramanujan theta function generalizes the form of the Jacobi theta functions, 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; it was his last major contribution to mathematics.

Definition

The Ramanujan theta function is defined as

:$f\left(a,b\right) = sum_\left\{n=-infty\right\}^infty a^\left\{n\left(n+1\right)/2\right\} ; b^\left\{n\left(n-1\right)/2\right\}$

for $|ab|<1.$ The Jacobi triple product identity then takes the form

:$f\left(a,b\right) = \left(-a; ab\right)_infty ;\left(-b; ab\right)_infty ;\left(ab;ab\right)_infty$

Here, the expression $\left(a;q\right)_n$ denotes the q-Pochhammer symbol. Identities that follow from this include

:$f\left(q,q\right) = sum_\left\{n=-infty\right\}^infty q^\left\{n^2\right\} = frac \left\{\left(-q;q^2\right)_infty \left(q^2;q^2\right)_infty\right\}\left\{\left(-q^2;q^2\right)_infty \left(q; q^2\right)_infty\right\}$

and

:$f\left(q,q^3\right) = sum_\left\{n=0\right\}^infty q^\left\{n\left(n+1\right)/2\right\} = frac \left\{\left(q^2;q^2\right)_infty\right\}\left\{\left(q; q^2\right)_infty\right\}$

and

:$f\left(-q,-q^2\right) = sum_\left\{n=-infty\right\}^infty \left(-1\right)^n q^\left\{n\left(3n-1\right)/2\right\} = \left(q;q\right)_infty$

this last being the Euler function, which is closely related to the Dedekind eta function.

References

* W.N. Bailey, "Generalized Hypergeometric Series", (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
* 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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