# §21.6 Products

## §21.6(i) Riemann Identity

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

 21.6.1 $\mathcal{K}={\mathbb{Z}^{g\times h}}\mathbf{T}/({\mathbb{Z}^{g\times h}}% \mathbf{T}\cap{\mathbb{Z}^{g\times h}}),$ ⓘ Defines: $\mathcal{K}$: set of matrices (locally) Symbols: $\mathbb{Z}$: set of all integers, $\cap$: intersection, $g$: positive integer and $h$: positive integer Permalink: http://dlmf.nist.gov/21.6.E1 Encodings: TeX, pMML, png See also: Annotations for §21.6(i), §21.6 and Ch.21

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

 21.6.2 $\mathcal{D}=|\mathbf{T}^{\mathrm{T}}{\mathbb{Z}^{h}}/(\mathbf{T}^{\mathrm{T}}{% \mathbb{Z}^{h}}\cap{\mathbb{Z}^{h}})|,$ ⓘ Defines: $\mathcal{D}$: number of elements (locally) Symbols: $\mathbb{Z}$: set of all integers, $\cap$: intersection and $h$: positive integer Permalink: http://dlmf.nist.gov/21.6.E2 Encodings: TeX, pMML, png See also: Annotations for §21.6(i), §21.6 and Ch.21

that is, $\mathcal{D}$ is the number of elements in the set containing all $h$-dimensional vectors obtained by multiplying $\mathbf{T}^{\mathrm{T}}$ 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

 21.6.3 $\prod_{j=1}^{h}\theta\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}\middle|% \boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}\in% \mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{2\pi i\operatorname{tr}\left[% \frac{1}{2}\mathbf{A}^{\mathrm{T}}\boldsymbol{{\Omega}}\mathbf{A}+\mathbf{A}^{% \mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\*\prod_{j=1}^{h}\theta\left(\mathbf% {z}_{j}+\boldsymbol{{\Omega}}\mathbf{a}_{j}+\mathbf{b}_{j}\middle|\boldsymbol{% {\Omega}}\right),$

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

 21.6.4 $\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\sum_{k=1}^{h}T_{jk}\mathbf{c}_{k% }}{\sum_{k=1}^{h}T_{jk}\mathbf{d}_{k}}\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}% \middle|\boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}% \in\mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{-2\pi i\sum_{j=1}^{h}\mathbf{% b}_{j}\cdot\mathbf{c}_{j}}\prod_{j=1}^{h}\theta\genfrac{[}{]}{0.0pt}{}{\mathbf% {a}_{j}+\mathbf{c}_{j}}{\mathbf{b}_{j}+\mathbf{d}_{j}}\left(\mathbf{z}_{j}% \middle|\boldsymbol{{\Omega}}\right),$

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

### Example

Let $h=4$ and

 21.6.5 $\mathbf{T}=\frac{1}{2}\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1\end{bmatrix}.$ ⓘ Referenced by: Erratum Equation (21.6.5) Permalink: http://dlmf.nist.gov/21.6.E5 Encodings: TeX, pMML, png Errata (effective with 1.0.17): Originally the prefactor $\frac{1}{2}$ on the right-hand side was missing. Reported 2017-08-12 by Wolfgang Bauhardt See also: Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

Then

 21.6.6 $\theta\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}}{2}\middle|% \boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}% -\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x% }-\mathbf{y}+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)% \theta\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}\middle|% \boldsymbol{{\Omega}}\right)=\frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in% \frac{1}{2}{\mathbb{Z}^{g}}/{\mathbb{Z}^{g}}}\,\sum_{\boldsymbol{{\beta}}\in% \frac{1}{2}{\mathbb{Z}^{g}}/{\mathbb{Z}^{g}}}e^{2\pi i\left(2\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+\boldsymbol{{% \alpha}}\cdot[\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}]\right)}\*\theta% \left(\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta% }}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{y}+\boldsymbol{{% \Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}% }\right)\theta\left(\mathbf{u}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{v}% +\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|% \boldsymbol{{\Omega}}\right),$

and

 21.6.7 $\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}+% \mathbf{c}_{3}+\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}+% \mathbf{d}_{3}+\mathbf{d}_{4}]}\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+% \mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{% }{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}-\mathbf{c}_{3}-\mathbf{c}_{4}]}{% \tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}-\mathbf{d}_{3}-\mathbf{d}_{4}]}% \left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol% {{\Omega}}\right)\*\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-% \mathbf{c}_{2}+\mathbf{c}_{3}-\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-% \mathbf{d}_{2}+\mathbf{d}_{3}-\mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y% }+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{% [}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-\mathbf{c}_{2}-\mathbf{c}_{3}+% \mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-\mathbf{d}_{2}-\mathbf{d}_{3}+% \mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}% \middle|\boldsymbol{{\Omega}}\right)\\ =\frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in\frac{1}{2}{\mathbb{Z}^{g}}/{% \mathbb{Z}^{g}}}\,\sum_{\boldsymbol{{\beta}}\in\frac{1}{2}{\mathbb{Z}^{g}}/{% \mathbb{Z}^{g}}}e^{-2\pi i\boldsymbol{{\beta}}\cdot[\mathbf{c}_{1}+\mathbf{c}_% {2}+\mathbf{c}_{3}+\mathbf{c}_{4}]}\*\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {1}+\boldsymbol{{\alpha}}}{\mathbf{d}_{1}+\boldsymbol{{\beta}}}\left(\mathbf{x% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {2}+\boldsymbol{{\alpha}}}{\mathbf{d}_{2}+\boldsymbol{{\beta}}}\left(\mathbf{y% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {3}+\boldsymbol{{\alpha}}}{\mathbf{d}_{3}+\boldsymbol{{\beta}}}\left(\mathbf{u% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {4}+\boldsymbol{{\alpha}}}{\mathbf{d}_{4}+\boldsymbol{{\beta}}}\left(\mathbf{v% }\middle|\boldsymbol{{\Omega}}\right).$

Let $\boldsymbol{{\alpha}}$, $\boldsymbol{{\beta}}$, $\boldsymbol{{\gamma}}$, $\boldsymbol{{\delta}}\in{\mathbb{R}^{g}}$. Then
 21.6.8 $\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\gamma}}}% \left(\mathbf{z}_{1}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0% .0pt}{}{\boldsymbol{{\beta}}}{\boldsymbol{{\delta}}}\left(\mathbf{z}_{2}% \middle|\boldsymbol{{\Omega}}\right)=\sum_{\boldsymbol{{\nu}}\in{\mathbb{Z}^{g% }}/\left(2{\mathbb{Z}^{g}}\right)}\theta\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[% \boldsymbol{{\alpha}}+\boldsymbol{{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{% \gamma}}+\boldsymbol{{\delta}}}\left(\mathbf{z}_{1}+\mathbf{z}_{2}\middle|2% \boldsymbol{{\Omega}}\right)\*\theta\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[% \boldsymbol{{\alpha}}-\boldsymbol{{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{% \gamma}}-\boldsymbol{{\delta}}}\left(\mathbf{z}_{1}-\mathbf{z}_{2}\middle|2% \boldsymbol{{\Omega}}\right).$
Thus $\boldsymbol{{\nu}}$ 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).