Anosov diffeomorphism

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.

Definition

The Ramanujan theta function is defined as

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

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

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

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

$f(q,q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}=(-q;q^2{)}_{\mathrm{\infty }}^2(q^2;q^2{)}_{\mathrm{\infty }}$

and

$f(q,q^3)=\sum _{n=0}^{\mathrm{\infty }}{q}^{n(n+1)/2}=(q^2;q^2{)}_{\mathrm{\infty }}(-q;q{)}_{\mathrm{\infty }}$

and

$f(-q,-q^2)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{n(3n-1)/2}=(q;q{)}_{\mathrm{\infty }}$

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:

$\vartheta (w,q)=f(q{w}^2,q{w}^{-2})$

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.
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