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 mathematician Srinivasa Ramanujan.
Contents
Definition
The Ramanujan theta function is defined as
f
(
a
,
b
)
=
∑
n
=
−
∞
∞
a
n
(
n
+
1
)
2
b
n
(
n
−
1
)
2
{\displaystyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{\frac {n(n+1)}{2}}\;b^{\frac {n(n-1)}{2}}}
for |ab| < 1. The Jacobi triple product identity then takes the form
f
(
a
,
b
)
=
(
−
a
;
a
b
)
∞
(
−
b
;
a
b
)
∞
(
a
b
;
a
b
)
∞
.
{\displaystyle f(a,b)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.}
Here, the expression
(
a
;
q
)
n
{\displaystyle (a;q)_{n}}
denotes the q-Pochhammer symbol. Identities that follow from this include
φ
(
q
)
=
f
(
q
,
q
)
=
∑
n
=
−
∞
∞
q
n
2
=
(
−
q
;
q
2
)
∞
2
(
q
2
;
q
2
)
∞
{\displaystyle \varphi (q)=f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={\left(-q;q^{2}\right)_{\infty }^{2}\left(q^{2};q^{2}\right)_{\infty }}}
and
ψ
(
q
)
=
f
(
q
,
q
3
)
=
∑
n
=
0
∞
q
n
(
n
+
1
)
2
=
(
q
2
;
q
2
)
∞
(
−
q
;
q
)
∞
{\displaystyle \psi (q)=f\left(q,q^{3}\right)=\sum _{n=0}^{\infty }q^{\frac {n(n+1)}{2}}={\left(q^{2};q^{2}\right)_{\infty }}{(-q;q)_{\infty }}}
and
f
(
−
q
)
=
f
(
−
q
,
−
q
2
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
n
(
3
n
−
1
)
2
=
(
q
;
q
)
∞
{\displaystyle f(-q)=f\left(-q,-q^{2}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{\frac {n(3n-1)}{2}}=(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:
ϑ
00
(
w
,
q
)
=
f
(
q
w
2
,
q
w
−
2
)
{\displaystyle \vartheta _{00}(w,q)=f\left(qw^{2},qw^{-2}\right)}
Integral representations
We have the following integral representation for the full two-parameter form of Ramanujan's theta function:
f
(
a
,
b
)
=
1
+
∫
0
∞
2
a
e
−
1
2
t
2
2
π
[
1
−
a
a
b
cosh
(
log
a
b
Application in string theory
The Ramanujan theta function is used to determine the critical dimensions in bosonic string theory, superstring theory and M-theory.