The root lattice 𝐴*_{𝑛} and Ramanujan’s circular summation of theta functions

Chua, Kok

doi:10.1090/S0002-9939-01-06080-4

The root lattice $A^*_n$ and Ramanujan’s circular summation of theta functions
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by Kok Seng Chua PDF

Proc. Amer. Math. Soc. 130 (2002), 1-8 Request permission

Abstract:

We relate a formula of Ramanujan on the circular summation of the $n$th power of theta functions, $F_n(q)$, to the theta series of the root lattice $A^*_n$. We then use properties of the lattice to show that $F_n$ includes an $\operatorname {SL}_2(\mathbf {Z})$ modular form when $n$ is an odd perfect square as well as to derive a very simple expression for $F_9(q)$.

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