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:

“The “female culture” has shifted more rapidly than the “male culture”; the image of the go-get ‘em woman has yet to be fully matched by the image of the let’s take-care-of-the-kids- together man. More important, over the last thirty years, men’s underlying feelings about taking responsibility at home have changed much less than women’s feelings have changed about forging some kind of identity at work.”
—Arlie Hochschild (20th century)