Ramanujan theta function

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This is not about the mock theta functions discovered by Ramanujan.

In mathematics, 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; it was his last major contribution to mathematics.

[edit] Definition

The Ramanujan theta function is defined as

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

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

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

f(q,q) = \sum_{n=-\infty}^\infty q^{n^2} = \frac {(-q;q^2)_\infty (q^2;q^2)_\infty} {(-q^2;q^2)_\infty (q; q^2)_\infty}

and

f(q,q^3) = \sum_{n=0}^\infty q^{n(n+1)/2} = \frac {(q^2;q^2)_\infty}{(q; q^2)_\infty}

and

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.

[edit] References

  • W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
  • George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.