In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan.
Definition The Ramanujan theta function is defined as
f ( a , b ) = ∑ n = − ∞ ∞ a n ( n + 1 ) 2 b n ( n − 1 ) 2 {\displaystyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{\frac {n(n+1)}{2}}\;b^{\frac {n(n-1)}{2}}}
for |ab| < 1. The Jacobi triple product identity then takes the form
f ( a , b ) = ( − a ; a b ) ∞ ( − b ; a b ) ∞ ( a b ; a b ) ∞ . {\displaystyle f(a,b)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.}
Here, the expression ( a ; q ) n {\displaystyle (a;q)_{n}} denotes the q-Pochhammer symbol. Identities that follow from this include
φ ( q ) = f ( q , q ) = ∑ n = − ∞ ∞ q n 2 = ( − q ; q 2 ) ∞ 2 ( q 2 ; q 2 ) ∞ {\displaystyle \varphi (q)=f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={\left(-q;q^{2}\right)_{\infty }^{2}\left(q^{2};q^{2}\right)_{\infty }}}
and
ψ ( q ) = f ( q , q 3 ) = ∑ n = 0 ∞ q n ( n + 1 ) 2 = ( q 2 ; q 2 ) ∞ ( − q ; q ) ∞ {\displaystyle \psi (q)=f\left(q,q^{3}\right)=\sum _{n=0}^{\infty }q^{\frac {n(n+1)}{2}}={\left(q^{2};q^{2}\right)_{\infty }}{(-q;q)_{\infty }}}
and
f ( − q ) = f ( − q , − q 2 ) = ∑ n = − ∞ ∞ ( − 1 ) n q n ( 3 n − 1 ) 2 = ( q ; q ) ∞ {\displaystyle f(-q)=f\left(-q,-q^{2}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{\frac {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:
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