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.


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



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 representations

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 OEIS: A000122 and OEIS: A010054 [2] also have the following integral representations:[1]

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

and that

Application in string theory

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


  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.
  • Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. 32. Cambridge: Cambridge University Press.
  • Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. 96 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-83357-4.
  • Hazewinkel, Michiel, ed. (2001) [1994], "Ramanujan function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press. ISBN 0-19-286189-1.
  • Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld.
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