Basic Hypergeometric Series - Ramanujan's Identity

Ramanujan's Identity

Ramanujan gave the identity

\;_1\psi_1 \left
= \sum_{n=-\infty}^\infty \frac {(a;q)_n} {(b;q)_n} z^n
= \frac {(b/a,q,q/az,az;q)_\infty }
{(b,b/az,q/a,z;q)_\infty}

valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for have been given by Bailey. Such identities can be understood to be generalizations of the Jacobi triple product theorem, which can be written using q-series as

\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n =
(q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.

Ken Ono gives a related formal power series

A(z;q) \stackrel{\rm{def}}{=} \frac{1}{1+z} \sum_{n=0}^\infty
\frac{(z;q)_n}{(-zq;q)_n}z^n =
\sum_{n=0}^\infty (-1)^n z^{2n} q^{n^2}.

Read more about this topic:  Basic Hypergeometric Series

Famous quotes containing the word identity:

    No other group in America has so had their identity socialized out of existence as have black women.... When black people are talked about the focus tends to be on black men; and when women are talked about the focus tends to be on white women.
    bell hooks (b. c. 1955)