# §20.7 Identities

## §20.7(i) Sums of Squares

 20.7.1 ${\theta_{3}}^{2}\left(0,q\right){\theta_{3}}^{2}\left(z,q\right)={\theta_{4}}^% {2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)+{\theta_{2}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome A&S Ref: 16.28.3 Permalink: http://dlmf.nist.gov/20.7.E1 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20
 20.7.2 ${\theta_{3}}^{2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)={\theta_{2}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{4}}^{2}\left(0,q% \right){\theta_{3}}^{2}\left(z,q\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome A&S Ref: 16.28.2 Permalink: http://dlmf.nist.gov/20.7.E2 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20
 20.7.3 ${\theta_{2}}^{2}\left(0,q\right){\theta_{4}}^{2}\left(z,q\right)={\theta_{3}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{4}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome A&S Ref: 16.28.1 Permalink: http://dlmf.nist.gov/20.7.E3 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20
 20.7.4 ${\theta_{2}}^{2}\left(0,q\right){\theta_{3}}^{2}\left(z,q\right)={\theta_{4}}^% {2}\left(0,q\right){\theta_{1}}^{2}\left(z,q\right)+{\theta_{3}}^{2}\left(0,q% \right){\theta_{2}}^{2}\left(z,q\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome A&S Ref: 16.28.4 Permalink: http://dlmf.nist.gov/20.7.E4 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20

Also

 20.7.5 ${\theta_{3}}^{4}\left(0,q\right)={\theta_{2}}^{4}\left(0,q\right)+{\theta_{4}}% ^{4}\left(0,q\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function and $q$: nome A&S Ref: 16.28.5 Referenced by: §23.6(ii) Permalink: http://dlmf.nist.gov/20.7.E5 Encodings: TeX, pMML, png See also: Annotations for §20.7(i), §20.7 and Ch.20

 20.7.6 ${\theta_{4}}^{2}\left(0,q\right)\theta_{1}\left(w+z,q\right)\theta_{1}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Referenced by: §20.7(ii) Permalink: http://dlmf.nist.gov/20.7.E6 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20
 20.7.7 ${\theta_{4}}^{2}\left(0,q\right)\theta_{2}\left(w+z,q\right)\theta_{2}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Permalink: http://dlmf.nist.gov/20.7.E7 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20
 20.7.8 ${\theta_{4}}^{2}\left(0,q\right)\theta_{3}\left(w+z,q\right)\theta_{3}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Permalink: http://dlmf.nist.gov/20.7.E8 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20
 20.7.9 ${\theta_{4}}^{2}\left(0,q\right)\theta_{4}\left(w+z,q\right)\theta_{4}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Referenced by: §20.7(ii) Permalink: http://dlmf.nist.gov/20.7.E9 Encodings: TeX, pMML, png See also: Annotations for §20.7(ii), §20.7 and Ch.20

For these and similar formulas see Lawden (1989, §1.4), Whittaker and Watson (1927, pp. 487–488), and Carlson (2011, §5).

Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$. The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012).

## §20.7(iii) Duplication Formula

 20.7.10 $\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z% ,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left% (0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}.$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Permalink: http://dlmf.nist.gov/20.7.E10 Encodings: TeX, pMML, png See also: Annotations for §20.7(iii), §20.7 and Ch.20

## §20.7(iv) Reduction Formulas for Products

 20.7.11 $\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z,q\right)}{\theta_{1}\left(2z% ,q^{2}\right)}=\frac{\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{% \theta_{4}\left(2z,q^{2}\right)}=\theta_{4}\left(0,q^{2}\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Referenced by: §20.7(iv), §20.7(ix) Permalink: http://dlmf.nist.gov/20.7.E11 Encodings: TeX, pMML, png See also: Annotations for §20.7(iv), §20.7 and Ch.20
 20.7.12 $\frac{\theta_{1}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}\right)}{\theta_{1}% \left(z,q\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}% \right)}{\theta_{2}\left(z,q\right)}=\tfrac{1}{2}\theta_{2}\left(0,q\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Referenced by: §20.7(iv), §20.7(ix) Permalink: http://dlmf.nist.gov/20.7.E12 Encodings: TeX, pMML, png See also: Annotations for §20.7(iv), §20.7 and Ch.20

Addendum: For a companion equation see (20.7.34).

## §20.7(v) Watson’s Identities

 20.7.13 $\theta_{1}\left(z,q\right)\theta_{1}\left(w,q\right)=\theta_{3}\left(z+w,q^{2}% \right)\theta_{2}\left(z-w,q^{2}\right)-\theta_{2}\left(z+w,q^{2}\right)\theta% _{3}\left(z-w,q^{2}\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Permalink: http://dlmf.nist.gov/20.7.E13 Encodings: TeX, pMML, png See also: Annotations for §20.7(v), §20.7 and Ch.20
 20.7.14 $\theta_{3}\left(z,q\right)\theta_{3}\left(w,q\right)=\theta_{3}\left(z+w,q^{2}% \right)\theta_{3}\left(z-w,q^{2}\right)+\theta_{2}\left(z+w,q^{2}\right)\theta% _{2}\left(z-w,q^{2}\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Permalink: http://dlmf.nist.gov/20.7.E14 Encodings: TeX, pMML, png See also: Annotations for §20.7(v), §20.7 and Ch.20

## §20.7(vi) Landen Transformations

With

 20.7.15 $A\equiv A(\tau)=\ifrac{1}{\theta_{4}\left(0\middle|2\tau\right)},$ ⓘ Defines: $A$ (locally) Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\equiv$: equals by definition and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.7.E15 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
 20.7.16 $\displaystyle\theta_{1}\left(2z\middle|2\tau\right)$ $\displaystyle=A\theta_{1}\left(z\middle|\tau\right)\theta_{2}\left(z\middle|% \tau\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $z$: complex, $\tau$: lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E16 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20 20.7.17 $\displaystyle\theta_{2}\left(2z\middle|2\tau\right)$ $\displaystyle=A\theta_{1}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{1}% \left(\tfrac{1}{4}\pi+z\middle|\tau\right),$ 20.7.18 $\displaystyle\theta_{3}\left(2z\middle|2\tau\right)$ $\displaystyle=A\theta_{3}\left(\tfrac{1}{4}\pi-z\middle|\tau\right)\theta_{3}% \left(\tfrac{1}{4}\pi+z\middle|\tau\right),$ 20.7.19 $\displaystyle\theta_{4}\left(2z\middle|2\tau\right)$ $\displaystyle=A\theta_{3}\left(z\middle|\tau\right)\theta_{4}\left(z\middle|% \tau\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $z$: complex, $\tau$: lattice parameter and $A$ Permalink: http://dlmf.nist.gov/20.7.E19 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20

Next, with

 20.7.20 $B\equiv B(\tau)=\ifrac{1}{\left(\theta_{3}\left(0\middle|\tau\right)\theta_{4}% \left(0\middle|\tau\right)\theta_{3}\left(\tfrac{1}{4}\pi\middle|\tau\right)% \right)},$ ⓘ Defines: $B$ (locally) Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.7.E20 Encodings: TeX, pMML, png See also: Annotations for §20.7(vi), §20.7 and Ch.20
 20.7.21 $\displaystyle\theta_{1}\left(4z\middle|4\tau\right)$ $\displaystyle=B\theta_{1}\left(z\middle|\tau\right)\theta_{1}\left(\tfrac{1}{4% }\pi-z\middle|\tau\right)\*\theta_{1}\left(\tfrac{1}{4}\pi+z\middle|\tau\right% )\theta_{2}\left(z\middle|\tau\right),$ 20.7.22 $\displaystyle\theta_{2}\left(4z\middle|4\tau\right)$ $\displaystyle=B\theta_{2}\left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{2}% \left(\tfrac{1}{8}\pi+z\middle|\tau\right)\*\theta_{2}\left(\tfrac{3}{8}\pi-z% \middle|\tau\right)\theta_{2}\left(\tfrac{3}{8}\pi+z\middle|\tau\right),$ 20.7.23 $\displaystyle\theta_{3}\left(4z\middle|4\tau\right)$ $\displaystyle=B\theta_{3}\left(\tfrac{1}{8}\pi-z\middle|\tau\right)\theta_{3}% \left(\tfrac{1}{8}\pi+z\middle|\tau\right)\*\theta_{3}\left(\tfrac{3}{8}\pi-z% \middle|\tau\right)\theta_{3}\left(\tfrac{3}{8}\pi+z\middle|\tau\right),$ 20.7.24 $\displaystyle\theta_{4}\left(4z\middle|4\tau\right)$ $\displaystyle=B\theta_{4}\left(z\middle|\tau\right)\theta_{4}\left(\tfrac{1}{4% }\pi-z\middle|\tau\right)\*\theta_{4}\left(\tfrac{1}{4}\pi+z\middle|\tau\right% )\theta_{3}\left(z\middle|\tau\right).$

## §20.7(vii) Derivatives of Ratios of Theta Functions

 20.7.25 $\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{\theta_{2}\left(z\middle|\tau\right)% }{\theta_{4}\left(z\middle|\tau\right)}\right)=-\frac{{\theta_{3}}^{2}\left(0% \middle|\tau\right)\theta_{1}\left(z\middle|\tau\right)\theta_{3}\left(z% \middle|\tau\right)}{{\theta_{4}}^{2}\left(z\middle|\tau\right)}.$

See Lawden (1989, pp. 19–20). This reference also gives the eleven additional identities for the permutations of the four theta functions.

## §20.7(viii) Transformations of Lattice Parameter

 20.7.26 $\displaystyle\theta_{1}\left(z\middle|\tau+1\right)$ $\displaystyle=e^{i\pi/4}\theta_{1}\left(z\middle|\tau\right),$ 20.7.27 $\displaystyle\theta_{2}\left(z\middle|\tau+1\right)$ $\displaystyle=e^{i\pi/4}\theta_{2}\left(z\middle|\tau\right),$ 20.7.28 $\displaystyle\theta_{3}\left(z\middle|\tau+1\right)$ $\displaystyle=\theta_{4}\left(z\middle|\tau\right),$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $z$: complex and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.7.E28 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20 20.7.29 $\displaystyle\theta_{4}\left(z\middle|\tau+1\right)$ $\displaystyle=\theta_{3}\left(z\middle|\tau\right).$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $z$: complex and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.7.E29 Encodings: TeX, pMML, png See also: Annotations for §20.7(viii), §20.7 and Ch.20

In the following equations $\tau^{\prime}=-1/\tau$, and all square roots assume their principal values.

 20.7.30 $\displaystyle(-i\tau)^{1/2}\theta_{1}\left(z\middle|\tau\right)$ $\displaystyle=-i\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{1}\left(z\tau^% {\prime}\middle|\tau^{\prime}\right),$ 20.7.31 $\displaystyle(-i\tau)^{1/2}\theta_{2}\left(z\middle|\tau\right)$ $\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{4}\left(z\tau^{% \prime}\middle|\tau^{\prime}\right),$ 20.7.32 $\displaystyle(-i\tau)^{1/2}\theta_{3}\left(z\middle|\tau\right)$ $\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{3}\left(z\tau^{% \prime}\middle|\tau^{\prime}\right),$ 20.7.33 $\displaystyle(-i\tau)^{1/2}\theta_{4}\left(z\middle|\tau\right)$ $\displaystyle=\exp\left(i\tau^{\prime}z^{2}/\pi\right)\theta_{2}\left(z\tau^{% \prime}\middle|\tau^{\prime}\right).$

These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183).

## §20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

 20.7.34 $\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}% \left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}% \right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^% {2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}.$ ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\mathrm{i}$: imaginary unit, $z$: complex and $q$: nome Referenced by: §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36 Permalink: http://dlmf.nist.gov/20.7.E34 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This equation has been added. See Walker (2012). See also: Annotations for §20.7(ix), §20.7 and Ch.20