Ramanujan Theta Function

In mathematics, particularly -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

:$f(a,b) = \sum_^\infty a^\frac \; b^\frac$

for . The Jacobi triple product identity then takes the form

:$f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.$

Here, the expression $(a;q)_n$ denotes the -Pochhammer symbol. Identities that follow from this include

:$\varphi(q) = f(q,q) = \sum_^\infty q^ =$

and

:$\psi(q) = f\left(q,q^3\right) = \sum_^\infty q^\frac =$

and

:$f(-q) = f\left(-q,-q^2\right) = \sum_^\infty (-1)^n q^\frac = (q;q)_\infty$

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:

:$\vartheta(w, q)=f\left(qw^2,qw^\right)$

Integral representations

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

:
f(a,b) = 1 + \int_0^ \frac\left
\frac
\rightdt +
\int_0^ \frac\left
\frac
\rightdt

The special cases of Ramanujan's theta functions given by and also have the following integral representations:

:
\begin
\varphi(q) & = 1 + \int_0^ \frac
\left frac
\rightdt \\ pt
\psi(q) & = \int_0^
\frac
\left frac
\rightdt
\end

This leads to several special case integrals for constants defined by these functions when (cf. theta function explicit values). In particular, we have that

:
\begin
\varphi\left(e^\right) & =
1 + \int_0^\infty \frac \left
\frac
\rightdt \\ pt
\frac & =
1 + \int_0^\infty \frac \left
\frac
\rightdt \\ pt
\frac \cdot
\frac & =
1 + \int_0^\infty \frac \left
\frac
\rightdt \\ pt
\frac \cdot
\frac & =
1 + \int_0^\infty \frac \left
\frac
\rightdt \\ pt
\frac \cdot
\frac & =
1 + \int_0^\infty \frac \left
\frac
\rightdt
\end

and that

:
\begin
\psi\left(e^\right) & =
\int_0^\infty \frac \left
\frac
\rightdt \\ pt
\frac \cdot
\frac & =
\int_0^\infty \frac \left
\frac
\rightdt \\ pt
\frac \cdot
\frac & =
\int_0^\infty \frac \left
\frac
\rightdt \\ pt
\frac \cdot
\frac & =
\int_0^\infty \frac \left
\frac
\rightdt
\end

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

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* {{Mathworld, RamanujanThetaFunctions, Ramanujan Theta Functions

Q-analogs
Elliptic functions
Theta functions
Srinivasa Ramanujan