§20.8 Watson’s Expansions

Keywords:: Watson’s expansions, theta functions
Permalink:: http://dlmf.nist.gov/20.8

20.8.1 $\frac{\mathop{\theta _{{2}}\/}\nolimits\!\left(0,q\right)\mathop{\theta _{{3}}\/}\nolimits\!\left(z,q\right)\mathop{\theta _{{4}}\/}\nolimits\!\left(z,q\right)}{\mathop{\theta _{{2}}\/}\nolimits\!\left(z,q\right)}=2\sum _{{n=-\infty}}^{{\infty}}\frac{(-1)^{n}q^{{n^{2}}}e^{{i2nz}}}{q^{{-n}}e^{{-iz}}+q^{n}e^{{iz}}}.$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $e$ : base of exponential function, $n$ : integer, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.8.E1
Encodings:: TeX, pMML, png

See Watson (1935a). This reference and Bellman (1961, pp. 46–47) include other expansions of this type.