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

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

for The Jacobi triple product identity then takes the form

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

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

:$f(q,q)\; =\; sum\_\{n=-infty\}^infty\; q^\{n^2\}\; =\; frac\; \{(-q;q^2)\_infty\; (q^2;q^2)\_infty\}\{(-q^2;q^2)\_infty\; (q;\; q^2)\_infty\}$

and

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

and

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

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.