§21.6 Products

Let $𝐓=[T_{jk}]$ be an arbitrary $h\times h$ orthogonal matrix (that is, $𝐓{𝐓}^{\mathrm{T}}=𝐈$) with rational elements. Also, let $𝐙$ be an arbitrary $g\times h$ matrix. Define

that is, $𝒦$ is the set of all $g\times h$ matrices that are obtained by premultiplying $𝐓$ by any $g\times h$ matrix with integer elements; two such matrices in $𝒦$ are considered equivalent if their difference is a matrix with integer elements. Also, let

that is, $𝒟$ is the number of elements in the set containing all $h$-dimensional vectors obtained by multiplying ${𝐓}^{\mathrm{T}}$ on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. Then

where ${𝐳}_j$, ${𝐚}_j$, ${𝐛}_j$ denote respectively the $j$th columns of $𝐙$, $𝐀$, $𝐁$. This is the Riemann identity. On using theta functions with characteristics, it becomes

where ${𝐜}_j$ and ${𝐝}_j$ are arbitrary $h$-dimensional vectors. Many identities involving products of theta functions can be established using these formulas.

Let $𝜶$, $𝜷$, $𝜸$, $𝜹\in {ℝ}^g$. Then

Thus $𝝂$ is a $g$-dimensional vector whose entries are either $0$ or $1$. For this result and a generalization see Koizumi (1976) and Belokolos et al. (1994, pp. 38–41). For addition formulas for classical theta functions see §20.7(ii).