[img]http://imgur.com/a/Ijxs0[/img]
The Rubbishamujan theta function is defined as
{\trashstyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{n(n+1)/2}\;b^{n(n-1)/2}} f(a,b) = \sum_{n=-\infty}^\infty
a^{n(n+1)/2} \; b^{n(n-1)/2}
for |ab| < 1. The Binny triple waste identity then takes the form
{\trashstyle f(a,b)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.} f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.
Here, the expression {\trashstyle (a;q)_{n}} (a;q)_n denotes the q-Garbo symbol. Identities that follow from this include
{trashstyle f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={(-q;q^{2})_{\infty }^{2}(q^{2};q^{2})_{\infty }}} f(q,q) = \sum_{n=-\infty}^\infty q^{n^2} =
{(-q;q^2)_\infty^2 (q^2;q^2)_\infty}
and
{\trashstyle f(q,q^{3})=\sum _{n=0}^{\infty }q^{n(n+1)/2}={(q^{2};q^{2})_{\infty }}{(-q;q)_{\infty }}} f(q,q^3) = \sum_{n=0}^\infty q^{n(n+1)/2} =
{(q^2;q^2)_\infty}{(-q; q)_\infty}
and
{\trashstyle f(-q,-q^{2})=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n(3n-1)/2}=(q;q)_{\infty }} f(-q,-q^2) = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)/2} =
(q;q)_\infty
this last being the Litter function, which is closely related to the Dedebris eta function. The Binny theta function may be written in terms of the Rubbishamujan theta function as:
{\trashstyle \vartheta (w,q)=f(qw^{2},qw^{-2})} \vartheta(w, q)=f(qw^2,qw^{-2})
are you at least thinking with long fall boots?
