§20.6 Power Series

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 $\sum _{{n=1}}^{\infty}(-1)^{{n-1}}n^{{-j}}=(1-2^{{1-j}})\sum _{{n=1}}^{\infty}n^{{-j}}$ .
Keywords:: power series, theta functions
Permalink:: http://dlmf.nist.gov/20.6

Assume

20.6.1 $\left|\pi z\right|<\min\left|z_{{m,n}}\right|,$

Symbols:: $m$ : integer, $n$ : integer and $z$ : complex
Permalink:: http://dlmf.nist.gov/20.6.E1
Encodings:: TeX, pMML, png

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

20.6.2 $\mathop{\theta _{{1}}\/}\nolimits\!\left(\pi z\middle|\tau\right)=\pi z{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0\middle|\tau\right)\mathop{\exp\/}\nolimits\!\left(-\sum _{{j=1}}^{{\infty}}\frac{1}{2j}\delta _{{2j}}(\tau)z^{{2j}}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : 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

20.6.3 $\mathop{\theta _{{2}}\/}\nolimits\!\left(\pi z\middle|\tau\right)=\mathop{\theta _{{2}}\/}\nolimits\!\left(0\middle|\tau\right)\mathop{\exp\/}\nolimits\!\left(-\sum _{{j=1}}^{{\infty}}\frac{1}{2j}\alpha _{{2j}}(\tau)z^{{2j}}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\alpha _{{2j}}(\tau)$ : coefficient
Permalink:: http://dlmf.nist.gov/20.6.E3
Encodings:: TeX, pMML, png

20.6.4 $\mathop{\theta _{{3}}\/}\nolimits\!\left(\pi z\middle|\tau\right)=\mathop{\theta _{{3}}\/}\nolimits\!\left(0\middle|\tau\right)\mathop{\exp\/}\nolimits\!\left(-\sum _{{j=1}}^{{\infty}}\frac{1}{2j}\beta _{{2j}}(\tau)z^{{2j}}\right),$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : complex, $\tau$ : lattice parameter and $\beta _{{2j}}(\tau)$ : coefficient
Permalink:: http://dlmf.nist.gov/20.6.E4
Encodings:: TeX, pMML, png

20.6.5 $\mathop{\theta _{{4}}\/}\nolimits\!\left(\pi z\middle|\tau\right)=\mathop{\theta _{{4}}\/}\nolimits\!\left(0\middle|\tau\right)\mathop{\exp\/}\nolimits\!\left(-\sum _{{j=1}}^{{\infty}}\frac{1}{2j}\gamma _{{2j}}(\tau)z^{{2j}}\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\exp\/}\nolimits z$ : 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

Here the coefficients are given by

20.6.6 $\delta _{{2j}}(\tau)=\left.\sum _{{n=-\infty}}^{{\infty}}\sum _{{\substack{m=-\infty\\ \left|m\right|+\left|n\right|\neq 0}}}^{{\infty}}\right.(m+n\tau)^{{-2j}},$

Symbols:: $m$ : integer, $n$ : integer, $\tau$ : lattice parameter and $\delta _{{2j}}(\tau)$ : coefficient
Referenced by:: §20.6, §20.6
Permalink:: http://dlmf.nist.gov/20.6.E6
Encodings:: TeX, pMML, png

20.6.7 $\alpha _{{2j}}(\tau)=\sum _{{n=-\infty}}^{{\infty}}\sum _{{m=-\infty}}^{{\infty}}(m-\tfrac{1}{2}+n\tau)^{{-2j}},$

Symbols:: $m$ : integer, $n$ : integer, $\tau$ : lattice parameter and $\alpha _{{2j}}(\tau)$ : coefficient
Permalink:: http://dlmf.nist.gov/20.6.E7
Encodings:: TeX, pMML, png

20.6.8 $\beta _{{2j}}(\tau)=\sum _{{n=-\infty}}^{{\infty}}\sum _{{m=-\infty}}^{{\infty}}(m-\tfrac{1}{2}+(n-\tfrac{1}{2})\tau)^{{-2j}},$

Symbols:: $m$ : integer, $n$ : integer, $\tau$ : lattice parameter and $\beta _{{2j}}(\tau)$ : coefficient
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.6.E8
Encodings:: TeX, pMML, png

20.6.9 $\gamma _{{2j}}(\tau)=\sum _{{n=-\infty}}^{{\infty}}\sum _{{m=-\infty}}^{{\infty}}(m+(n-\tfrac{1}{2})\tau)^{{-2j}},$

Symbols:: $m$ : integer, $n$ : integer, $\tau$ : lattice parameter and $\gamma _{{2j}}(\tau)$ : coefficient
Permalink:: http://dlmf.nist.gov/20.6.E9
Encodings:: TeX, pMML, png

and satisfy

20.6.10

$\alpha _{{2j}}(\tau)=2^{{2j}}\delta _{{2j}}(2\tau)-\delta _{{2j}}(\tau),$

$\beta _{{2j}}(\tau)=2^{{2j}}\gamma _{{2j}}(2\tau)-\gamma _{{2j}}(\tau).$

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

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 $n\geq 2$ .