§21.6 Products

Keywords:: Riemann theta functions
Referenced by:: §20.11(iv)
Permalink:: http://dlmf.nist.gov/21.6

§21.6(i) Riemann Identity
§21.6(ii) Addition Formulas

§21.6(i) Riemann Identity

Notes:: See Mumford (1983, pp. 211–216).
Keywords:: Riemann identity, Riemann theta functions, Riemann theta functions with characteristics, matrix, vector
Permalink:: http://dlmf.nist.gov/21.6.i

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 $\mathcal{K}=\Integer^{{g\times h}}\mathbf{T}/(\Integer^{{g\times h}}\mathbf{T}\cap\Integer^{{g\times h}}),$

Symbols:: $\Integer$ : set of all integers, $\cap$ : intersection, $g$ : positive integer, $h$ : positive integer and $\mathcal{K}$ : set of matrices
Permalink:: http://dlmf.nist.gov/21.6.E1
Encodings:: TeX, pMML, png

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 $\mathcal{D}=|\mathbf{T}^{{\mathrm{T}}}\Integer^{h}/(\mathbf{T}^{{\mathrm{T}}}\Integer^{h}\cap\Integer^{h})|,$

Symbols:: $\Integer$ : set of all integers, $\cap$ : intersection, $h$ : positive integer and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E2
Encodings:: TeX, pMML, png

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 $\prod _{{j=1}}^{h}\mathop{\theta\/}\nolimits\!\left(\sum _{{k=1}}^{h}T_{{jk}}\mathbf{z}_{k}\middle|\boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum _{{\mathbf{A}\in\mathcal{K}}}\sum _{{\mathbf{B}\in\mathcal{K}}}e^{{2\pi i\trace\left[\frac{1}{2}\mathbf{A}^{{\mathrm{T}}}\boldsymbol{{\Omega}}\mathbf{A}+\mathbf{A}^{{\mathrm{T}}}[\mathbf{Z}+\mathbf{B}]\right]}}\*\prod _{{j=1}}^{h}\mathop{\theta\/}\nolimits\!\left(\mathbf{z}_{j}+\boldsymbol{{\Omega}}\mathbf{a}_{j}+\mathbf{b}_{j}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\in$ : element of, $e$ : base of exponential function, $\trace\mathbf{A}$ : trace of matrix $\mathbf{A}$ , $g$ : positive integer, $h$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $T_{{jk}}$ : element, $\mathcal{K}$ : set of matrices and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E3
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, $e$ : base of exponential function, $g$ : positive integer, $h$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $T_{{jk}}$ : element, $\mathcal{K}$ : set of matrices and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E4
Encodings:: TeX, pMML, png

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 $\mathbf{T}=\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1\end{bmatrix}.$

Permalink:: http://dlmf.nist.gov/21.6.E5
Encodings:: TeX, pMML, png

Then

21.6.6 $\mathop{\theta\/}\nolimits\!\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}\nolimits\!\left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}\nolimits\!\left(\frac{\mathbf{x}-\mathbf{y}+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}\nolimits\!\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)=\frac{1}{2^{g}}\sum _{{\boldsymbol{{\alpha}}\in\frac{1}{2}\Integer^{g}/\Integer^{g}}}\,\sum _{{\boldsymbol{{\beta}}\in\frac{1}{2}\Integer^{g}/\Integer^{g}}}e^{{2\pi i\left(2\boldsymbol{{\alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+\boldsymbol{{\alpha}}\cdot[\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}]\right)}}\*\mathop{\theta\/}\nolimits\!\left(\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}\nolimits\!\left(\mathbf{y}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}\nolimits\!\left(\mathbf{u}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}\nolimits\!\left(\mathbf{v}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\in$ : element of, $e$ : base of exponential function, $\Integer$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E6
Encodings:: TeX, pMML, png

and

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

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, $e$ : base of exponential function, $\Integer$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E7
Encodings:: TeX, pMML, png

§21.6(ii) Addition Formulas

Notes:: See Dubrovin (1981, pp. 22–23).
Keywords:: Riemann theta functions with characteristics
Permalink:: http://dlmf.nist.gov/21.6.ii

Let $\boldsymbol{{\alpha}}$ , $\boldsymbol{{\beta}}$ , $\boldsymbol{{\gamma}}$ , $\boldsymbol{{\delta}}\in\Real^{g}$ . Then

21.6.8 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\gamma}}}\/}\nolimits\!\left(\mathbf{z}_{1}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\beta}}}{\boldsymbol{{\delta}}}\/}\nolimits\!\left(\mathbf{z}_{2}\middle|\boldsymbol{{\Omega}}\right)=\sum _{{\boldsymbol{{\nu}}\in\Integer^{g}/\left(2\Integer^{g}\right)}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{\gamma}}+\boldsymbol{{\delta}}}\/}\nolimits\!\left(\mathbf{z}_{1}+\mathbf{z}_{2}\middle|2\boldsymbol{{\Omega}}\right)\*\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[\boldsymbol{{\alpha}}-\boldsymbol{{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{\gamma}}-\boldsymbol{{\delta}}}\/}\nolimits\!\left(\mathbf{z}_{1}-\mathbf{z}_{2}\middle|2\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, $\Integer$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $\boldsymbol{{\delta}}$ : $g$ -dimensional vector and $\boldsymbol{{\gamma}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E8
Encodings:: TeX, pMML, png

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).

§21.6 Products

Contents

§21.6(i) Riemann Identity

¶ Example

§21.6(ii) Addition Formulas