20 Theta Functions Properties 20.5 Infinite Products and Related Results 20.7 Identities

§20.6 Power Series

ⓘ

Keywords:: power series, theta functions
Notes:: See Walker (1996, §3.2) and §23.9. (20.6.2)–(20.6.5) may be derived by termwise expansion in (20.5.14)–(20.5.17). (20.6.10) may be derived from (20.6.6) and (20.6.8) by subtraction of terms with even $j$ , in a similar manner to $\sum_{n=1}^{\infty}(-1)^{n-1}n^{-j}=(1-2^{1-j})\sum_{n=1}^{\infty}n^{-j}$ .
Permalink:: http://dlmf.nist.gov/20.6
See also:: Annotations for Ch.20

Assume

20.6.1		$\left\|\pi z\right\|<\min\left\|z_{m,n}\right\|,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $n$ : integer and $z$ : complex Permalink: http://dlmf.nist.gov/20.6.E1 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20

where $z_{m,n}$ is given by (20.2.5) and the minimum is for $m,n\in\mathbb{Z}$ , except $m=n=0$ . Then

20.6.2	$\displaystyle\theta_{1}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\pi z\theta_{1}'\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^% {\infty}\frac{1}{2j}\delta_{2j}(\tau)z^{2j}\right),$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\delta_{2j}(\tau)$ : coefficient Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E2 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.3	$\displaystyle\theta_{2}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\theta_{2}\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\alpha_{2j}(\tau)z^{2j}\right),$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\alpha_{2j}(\tau)$ : coefficient Permalink: http://dlmf.nist.gov/20.6.E3 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.4	$\displaystyle\theta_{3}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\theta_{3}\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\beta_{2j}(\tau)z^{2j}\right),$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\beta_{2j}(\tau)$ : coefficient Permalink: http://dlmf.nist.gov/20.6.E4 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.5	$\displaystyle\theta_{4}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\theta_{4}\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\gamma_{2j}(\tau)z^{2j}\right).$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\gamma_{2j}(\tau)$ : coefficient Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E5 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20

Here the coefficients are given by

20.6.6	$\displaystyle\delta_{2j}(\tau)$	$\displaystyle=\left.\sum_{n=-\infty}^{\infty}\sum_{\begin{subarray}{c}m=-% \infty\\ \left\|m\right\|+\left\|n\right\|\neq 0\end{subarray}}^{\infty}\right.(m+n\tau)^{-% 2j},$
ⓘ Defines: $\delta_{2j}(\tau)$ : coefficient (locally) Symbols: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter Referenced by: §20.6, §20.6 Permalink: http://dlmf.nist.gov/20.6.E6 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.7	$\displaystyle\alpha_{2j}(\tau)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{1}{2% }+n\tau)^{-2j},$
ⓘ Defines: $\alpha_{2j}(\tau)$ : coefficient (locally) Symbols: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.6.E7 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.8	$\displaystyle\beta_{2j}(\tau)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{1}{2% }+(n-\tfrac{1}{2})\tau)^{-2j},$
ⓘ Defines: $\beta_{2j}(\tau)$ : coefficient (locally) Symbols: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E8 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.9	$\displaystyle\gamma_{2j}(\tau)$	$\displaystyle=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m+(n-\tfrac{1% }{2})\tau)^{-2j},$
ⓘ Defines: $\gamma_{2j}(\tau)$ : coefficient (locally) Symbols: $m$ : integer, $n$ : integer and $\tau$ : lattice parameter Permalink: http://dlmf.nist.gov/20.6.E9 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20

and satisfy

20.6.10	$\displaystyle\alpha_{2j}(\tau)$	$\displaystyle=2^{2j}\delta_{2j}(2\tau)-\delta_{2j}(\tau),$
20.6.10	$\displaystyle\beta_{2j}(\tau)$	$\displaystyle=2^{2j}\gamma_{2j}(2\tau)-\gamma_{2j}(\tau).$
ⓘ Symbols: $\tau$ : lattice parameter, $\delta_{2j}(\tau)$ : coefficient, $\alpha_{2j}(\tau)$ : coefficient, $\beta_{2j}(\tau)$ : coefficient and $\gamma_{2j}(\tau)$ : coefficient Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §20.6 and Ch.20

In the double series the order of summation is important only when $j=1$ . For further information on $\delta_{2j}$ see §23.9: since the double sums in (20.6.6) and (23.9.1) are the same, we have $\delta_{2n}=c_{n}/(2n-1)$ when $n\geq 2$ .

20.5 Infinite Products and Related Results 20.7 Identities

NIST

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20.6.2	$\displaystyle\theta_{1}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\pi z\theta_{1}'\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^% {\infty}\frac{1}{2j}\delta_{2j}(\tau)z^{2j}\right),$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\delta_{2j}(\tau)$ : coefficient Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E2 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.3	$\displaystyle\theta_{2}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\theta_{2}\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\alpha_{2j}(\tau)z^{2j}\right),$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\alpha_{2j}(\tau)$ : coefficient Permalink: http://dlmf.nist.gov/20.6.E3 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.4	$\displaystyle\theta_{3}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\theta_{3}\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\beta_{2j}(\tau)z^{2j}\right),$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\beta_{2j}(\tau)$ : coefficient Permalink: http://dlmf.nist.gov/20.6.E4 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20
20.6.5	$\displaystyle\theta_{4}\left(\pi z\middle\|\tau\right)$	$\displaystyle=\theta_{4}\left(0\middle\|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\gamma_{2j}(\tau)z^{2j}\right).$
ⓘ 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, $\exp\NVar{z}$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\gamma_{2j}(\tau)$ : coefficient Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E5 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20