Ramanujan theta function

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 Srinivasa Ramanujan.

Definition

The Ramanujan theta function is defined as

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

for |ab| < 1. The Jacobi triple product identity then takes the form

${\displaystyle f(a,b)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.}$

Here, the expression ${\displaystyle (a;q)_{n}}$ denotes the q-Pochhammer symbol. Identities that follow from this include

${\displaystyle \varphi (q)=f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={(-q;q^{2})_{\infty }^{2}(q^{2};q^{2})_{\infty }}}$

and

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

and

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

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:

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

Integral representations

We have the following integral representation for the full two-parameter form of Ramanujan's theta function:[1]

${\displaystyle {\begin{aligned}f(a,b)&=1+\int _{0}^{\infty }{\frac {2ae^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {1-a{\sqrt {ab}}\cosh \left({\sqrt {\log(ab)}}t\right)}{a^{3}b-2a{\sqrt {ab}}\cosh \left({\sqrt {\log(ab)}}t\right)+1}}\right]dt+\int _{0}^{\infty }{\frac {2be^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {1-b{\sqrt {ab}}\cosh \left({\sqrt {\log(ab)}}t\right)}{ab^{3}-2b{\sqrt {ab}}\cosh \left({\sqrt {\log(ab)}}t\right)+1}}\right]dt.\end{aligned}}}$

The special cases of Ramanujan's theta functions given by ${\displaystyle \varphi (q):=f(q,q)}$ OEIS: A000122 and ${\displaystyle \psi (q):=f(q,q^{3})}$ OEIS: A010054 [2] also have the following integral representations:[1]

${\displaystyle {\begin{aligned}\varphi (q)&=1+\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {4q\left(1-q^{2}\cosh \left({\sqrt {2\log(q)}}t\right)\right)}{q^{4}-2q^{2}\cosh \left({\sqrt {2\log(q)}}t\right)+1}}\right]dt\\\psi (q)&=\int _{0}^{\infty }{\frac {2e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {\left(1-{\sqrt {q}}\cosh \left({\sqrt {\log(q)}}t\right)\right)}{q-2{\sqrt {q}}\cosh \left({\sqrt {\log(q)}}t\right)+1}}\right]dt.\end{aligned}}}$

This leads to several special case integrals for constants defined by these functions when ${\displaystyle q:=\exp(-k\pi )}$ (cf. theta function explicit values). In particular, we have that [1]

${\displaystyle {\begin{aligned}\varphi \left(e^{-k\pi }\right)&=1+\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {4e^{k\pi }\left(e^{2k\pi }-\cos \left({\sqrt {2\pi k}}t\right)\right)}{e^{4k\pi }-2e^{2k\pi }\cos \left({\sqrt {2\pi k}}t\right)+1}}\right]dt\\{\frac {\pi ^{1/4}}{\Gamma \left({\frac {3}{4}}\right)}}&=1+\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {4e^{\pi }\left(e^{2\pi }-\cos \left({\sqrt {2\pi }}t\right)\right)}{e^{4\pi }-2e^{2\pi }\cos \left({\sqrt {2\pi }}t\right)+1}}\right]dt\\{\frac {\pi ^{1/4}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {{\sqrt {2}}+2}}{2}}&=1+\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {4e^{2\pi }\left(e^{4\pi }-\cos \left(2{\sqrt {\pi }}t\right)\right)}{e^{8\pi }-2e^{4\pi }\cos \left(2{\sqrt {\pi }}t\right)+1}}\right]dt\\{\frac {\pi ^{1/4}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {{\sqrt {3}}+1}}{2^{1/4}3^{3/8}}}&=1+\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {4e^{3\pi }\left(e^{6\pi }-\cos \left({\sqrt {6\pi }}t\right)\right)}{e^{12\pi }-2e^{6\pi }\cos \left({\sqrt {6\pi }}t\right)+1}}\right]dt\\{\frac {\pi ^{1/4}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {5+2{\sqrt {5}}}}{5^{3/4}}}&=1+\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {4e^{5\pi }\left(e^{10\pi }-\cos \left({\sqrt {10\pi }}t\right)\right)}{e^{20\pi }-2e^{10\pi }\cos \left({\sqrt {10\pi }}t\right)+1}}\right]dt.\end{aligned}}}$

and that

${\displaystyle {\begin{aligned}\psi \left(e^{-k\pi }\right)&=\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {k\pi }}t\right)-e^{k\pi /2}}{\cos \left({\sqrt {k\pi }}t\right)-\cosh \left({\frac {k\pi }{2}}\right)}}\right]dt\\{\frac {\pi ^{1/4}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\pi /8}}{2^{5/8}}}&=\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\pi }}t\right)-e^{\pi /2}}{\cos \left({\sqrt {\pi }}t\right)-\cosh \left({\frac {\pi }{2}}\right)}}\right]dt\\{\frac {\pi ^{1/4}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\pi /4}}{2^{5/4}}}&=\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {2\pi }}t\right)-e^{\pi }}{\cos \left({\sqrt {2\pi }}t\right)-\cosh \left(\pi \right)}}\right]dt\\{\frac {\pi ^{1/4}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\left({\sqrt {2}}+1\right)^{1/4}e^{\pi /16}}{2^{7/16}}}&=\int _{0}^{\infty }{\frac {e^{-t^{2}/2}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\frac {\pi }{2}}}t\right)-e^{\pi /4}}{\cos \left({\sqrt {\frac {\pi }{2}}}t\right)-\cosh \left({\frac {\pi }{4}}\right)}}\right]dt.\end{aligned}}}$

Application in string theory

The Ramanujan theta function is used to determine the critical dimensions in Bosonic string theory, superstring theory and M-theory.

References

1. Schmidt, M. D. (2017). "Square series generating function transformations" (PDF). Journal of Inequalities and Special Functions. 8 (2).
2. Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld. Retrieved 29 April 2018.

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