- Ramanujan theta function
In
mathematics , the Ramanujan theta function generalizes the form of the Jacobitheta function s, while capturing their general properties. In particular, theJacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named afterSrinivasa 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 ab|<1. 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 theDedekind 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.
Wikimedia Foundation. 2010.