Ramanujan theta function

Not to be confused with the mock theta functions discovered by Ramanujan.

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

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

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

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

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

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

and

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

and

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

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:

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

Integral representations

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

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

The special cases of Ramanujan's theta functions given by φ(q) := f(q, q) OEIS:  A000122 and ψ(q) := f(q, q³) OEIS:  A010054 ^[2] also have the following integral representations: ^[1]

${\displaystyle {\begin{aligned}\varphi (q)&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4q\left(1-q^{2}\cosh \left({\sqrt {2\log q}}\,t\right)\right)}{q^{4}-2q^{2}\cosh \left({\sqrt {2\log q}}\,t\right)+1}}\right]dt\\[6pt]\psi (q)&=\int _{0}^{\infty }{\frac {2e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {1-{\sqrt {q}}\cosh \left({\sqrt {\log q}}\,t\right)}{q-2{\sqrt {q}}\cosh \left({\sqrt {\log q}}\,t\right)+1}}\right]dt\end{aligned}}}$

This leads to several special case integrals for constants defined by these functions when q := e^−kπ (cf. theta function explicit values). In particular, we have that ^[1]

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

and that

${\displaystyle {\begin{aligned}\psi \left(e^{-k\pi }\right)&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {k\pi }}\,t\right)-e^{\frac {k\pi }{2}}}{\cos \left({\sqrt {k\pi }}\,t\right)-\cosh {\frac {k\pi }{2}}}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\frac {\pi }{8}}}{2^{\frac {5}{8}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\pi }}\,t\right)-e^{\frac {\pi }{2}}}{\cos \left({\sqrt {\pi }}\,t\right)-\cosh {\frac {\pi }{2}}}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\frac {\pi }{4}}}{2^{\frac {5}{4}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {2\pi }}\,t\right)-e^{\pi }}{\cos \left({\sqrt {2\pi }}\,t\right)-\cosh \pi }}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {{\sqrt[{4}]{1+{\sqrt {2}}}}\,e^{\frac {\pi }{16}}}{2^{\frac {7}{16}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\frac {\pi }{2}}}\,t\right)-e^{\frac {\pi }{4}}}{\cos \left({\sqrt {\frac {\pi }{2}}}\,t\right)-\cosh {\frac {\pi }{4}}}}\right]dt\end{aligned}}}$

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). arXiv: 1609.02803 .
2. Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld. Retrieved 29 April 2018.

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