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The Ramanujan theta function is defined as
{\displaystyle 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 Jacobi triple product identity then takes the form
{\displaystyle 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 {\displaystyle (a;q)_{n}} (a;q)_n denotes the q-Pochhammer symbol. Identities that follow from this include
{\displaystyle 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
{\displaystyle 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
{\displaystyle 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 Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
{\displaystyle \vartheta (w,q)=f(qw^{2},qw^{-2})} \vartheta(w, q)=f(qw^2,qw^{-2})
The Ramanujan theta function is defined as
{\displaystyle 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 Jacobi triple product identity then takes the form
{\displaystyle 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 {\displaystyle (a;q)_{n}} (a;q)_n denotes the q-Pochhammer symbol. Identities that follow from this include
{\displaystyle 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
{\displaystyle 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
{\displaystyle 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 Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
{\displaystyle \vartheta (w,q)=f(qw^{2},qw^{-2})} \vartheta(w, q)=f(qw^2,qw^{-2})
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