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Ramanujan theta function

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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.

Contents

Definition

The Ramanujan theta function is defined as

f ( a , b ) = ∑ n = − ∞ ∞ 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 ; a b ) ∞ ( − b ; a b ) ∞ ( a b ; a b ) ∞ .

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

f ( q , q ) = ∑ n = − ∞ ∞ q n 2 = ( − q ; q 2 ) ∞ 2 ( q 2 ; q 2 ) ∞

and

f ( q , q 3 ) = ∑ n = 0 ∞ q n ( n + 1 ) / 2 = ( q 2 ; q 2 ) ∞ ( − q ; q ) ∞

and

f ( − q , − q 2 ) = ∑ n = − ∞ ∞ ( − 1 ) n q n ( 3 n − 1 ) / 2 = ( q ; q ) ∞

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 ( w , q ) = f ( q w 2 , q w − 2 )

Application in string theory

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

References

Ramanujan theta function Wikipedia
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