Ramanujan theta function

The Ramanujan theta function according to S. Ramanujan is through

${\displaystyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{n(n+1)/2}\;b^{n(n-1)/2}}$

given with . ${\displaystyle |ab|<1}$

The following applies (the function is symmetrical in the two variables). For the first terms we get: ${\displaystyle f(a,b)=f(b,a)}$

${\displaystyle f(a,b)=1+a+b+ab(a^{2}+b^{2})+{(ab)}^{3}(a^{3}+b^{3})+{(ab)}^{6}(a^{4}+b^{4})+\cdot \cdot \cdot }$

With the q-Pochhammer symbol , Ramanujan's theta function is expressed as follows: ${\displaystyle (a;q)_{n}}$

${\displaystyle f(a,b)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }=\prod _{m=0}^{\infty }\left(1+a(ab)^{m}\right)\,\left(1+b(ab)^{m}\right)\,\left(1-(ab)^{m+1}\right)}$

which is equivalent to the Jacobi triple product . For the special case

${\displaystyle f(-q,-q^{2})=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n(3n-1)/2}=(q;q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}$

the Jacobi triple product gives the pentagonal number theorem . Sometimes people write. The function is closely related to Dedekind's η-function and its reciprocal is the generating function for partitions . ${\displaystyle f(-q,-q^{2})=f(-q)}$

Further special cases are the Ramanujan function: ${\displaystyle \varphi }$

${\displaystyle \varphi (q)=f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={\frac {(-q;q^{2})_{\infty }(q^{2};q^{2})_{\infty }}{(-q^{2};q^{2})_{\infty }(q;q^{2})_{\infty }}}={\frac {(-q;-q)_{\infty }}{(q;-q)_{\infty }}}=1+2q+2q^{4}+2q^{9}+2q^{16}+2q^{25}+....}$

and Ramanujan's function: ${\displaystyle \psi }$

${\displaystyle \psi (q)=f(q,q^{3})=\sum _{n=0}^{\infty }q^{n(n+1)/2}={\frac {(q^{2};q^{2})_{\infty }}{(q;q^{2})_{\infty }}}}$

The Jacobian theta function results as:

${\displaystyle \vartheta (w,q)=f(qw^{2},qw^{-2})=\sum _{n=-\infty }^{\infty }q^{n^{2}}\,w^{2n}}$

with , so that the usual representation results: ${\displaystyle q=e^{\pi i\tau }}$ ${\displaystyle w=e^{\pi iz}}$

${\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz)}$

Web links

• Ramanujan Theta Function, Mathworld