On an identity of theta functions obtained from weight enumerators of linear codes (Q1281659)

The following formula is proved: \[ \theta_{A^*_{n-1}}(n\tau)= F (\tau) \] for any integer \(n>1\), where on the left hand side we have the theta function of the dual \(A^*_{n-1}\) of the root lattice \(A_{n-1}\) and where \[ F(\tau)=\sum^\infty_{m=0}a_m\exp(\pi im\tau) \] with \(q=1\), \(a_j=0\) \((0<j<n-1)\), \(a_{n-1} =2n\). This generalizes a result of \textit{S. S. Rangachari} [J. Number Theory 48, 364-372 (1994; Zbl 0813.11030)].