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.

DefinitionEdit

The Ramanujan theta function is defined as

 

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

 

Here, the expression   denotes the q-Pochhammer symbol. Identities that follow from this include

 

and

 

and

 

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:

 

Integral representationsEdit

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

 

The special cases of Ramanujan's theta functions given by φ(q) := f(q, q) OEISA000122 and ψ(q) := f(q, q3) OEISA010054 [2] also have the following integral representations:[1]

 

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

 

and that

 

Application in string theoryEdit

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

ReferencesEdit

  1. ^ a b c 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.