# §28.14 Fourier Series

The Fourier series

 28.14.1 $\displaystyle\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)e^{i(\nu+2m)z},$ 28.14.2 $\displaystyle\mathop{\mathrm{ce}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)\mathop{\cos\/}\nolimits% (\nu+2m)z,$ 28.14.3 $\displaystyle\mathop{\mathrm{se}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\sum_{m=-\infty}^{\infty}c^{\nu}_{2m}(q)\mathop{\sin\/}\nolimits% (\nu+2m)z,$

converge absolutely and uniformly on all compact sets in the $z$-plane. The coefficients satisfy

 28.14.4 ${qc_{2m+2}-\left(a-(\nu+2m)^{2}\right)c_{2m}+qc_{2m-2}=0},$ $a=\mathop{\lambda_{\nu}\/}\nolimits\!\left(q\right),c_{2m}=c_{2m}^{\nu}(q)$,

and the normalization relation

 28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$

compare (28.12.5). Ambiguities in sign are resolved by (28.14.9) when $q=0$, and by continuity for other values of $q$.

The rate of convergence is indicated by

 28.14.6 $\frac{c^{\nu}_{2m}(q)}{c^{\nu}_{2m\mp 2}(q)}=\frac{-q}{4m^{2}}\left(1+\mathop{% O\/}\nolimits\!\left(\frac{1}{m}\right)\right),$ $m\to\pm\infty$.

For changes of sign of $\nu$, $q$, and $m$,

 28.14.7 $\displaystyle c_{-2m}^{-\nu}(q)$ $\displaystyle=c_{2m}^{\nu}(q),$ 28.14.8 $\displaystyle c_{2m}^{\nu}(-q)$ $\displaystyle=(-1)^{m}c_{2m}^{\nu}(q).$

When $q=0$,

 28.14.9 $\displaystyle c_{0}^{\nu}(0)$ $\displaystyle=1,$ $\displaystyle c_{2m}^{\nu}(0)$ $\displaystyle=0,$ $m\neq 0$. Symbols: $m$: integer, $\nu$: complex parameter and $c_{2m}(q)$: coefficients A&S Ref: 20.8.2 Referenced by: §28.14 Permalink: http://dlmf.nist.gov/28.14.E9 Encodings: TeX, TeX, pMML, pMML, png, png

When $q\to 0$ with $m$ ($\geq 1$) and $\nu$ fixed,

 28.14.10 $c_{2m}^{\nu}(q)=\left(\frac{(-1)^{m}q^{m}\mathop{\Gamma\/}\nolimits\!\left(\nu% +1\right)}{m!\,2^{2m}\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}+\mathop% {O\/}\nolimits\!\left(q^{m+2}\right)\right)c_{0}^{\nu}(q).$