# §29.6 Fourier Series

## §29.6(i) Function $\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$

With $\phi=\frac{1}{2}\pi-\mathop{\mathrm{am}\/}\nolimits\left(z,k\right)$, as in (29.2.5), we have

 29.6.1 $\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\tfrac{1}{2}A% _{0}+\sum_{p=1}^{\infty}A_{2p}\mathop{\cos\/}\nolimits\!\left(2p\phi\right).$

Here

 29.6.2 $H=2\!\mathop{a^{2m}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2},$ Defines: $H$ (locally) Symbols: $\mathop{a^{n}_{\nu}\/}\nolimits\!\left(k^{2}\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E2 Encodings: TeX, pMML, png
 29.6.3 $(\beta_{0}-H)A_{0}+\alpha_{0}A_{2}=0,$ Symbols: $\alpha_{p}$, $\beta_{p}$, $H$ and $A_{2p}$: coefficients Referenced by: §29.15(i), §29.20(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E3 Encodings: TeX, pMML, png
 29.6.4 $\gamma_{p}A_{2p-2}+(\beta_{p}-H)A_{2p}+\alpha_{p}A_{2p+2}=0,$ $p\geq 1$,

with $\alpha_{p}$, $\beta_{p}$, and $\gamma_{p}$ as in (29.3.11) and (29.3.12), and

 29.6.5 $\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2}=1,$ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Referenced by: §29.20(i), §29.6(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E5 Encodings: TeX, pMML, png
 29.6.6 $\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p}>0.$ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Referenced by: §29.20(i), §29.6(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E6 Encodings: TeX, pMML, png

When $\nu\neq 2n$, where $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ the system (29.6.4)–(29.6.6) has a unique recessive solution $A_{0},A_{2},A_{4},\dots$; furthermore

 29.6.7 $\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n$, or $\nu=2n$ and $m>n$.

In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies.

In the special case $\nu=2n$, $m=0,1,\dots,n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\dots$. This solution can be constructed from (29.6.4) by backward recursion, starting with $A_{2n+2}=0$ and an arbitrary nonzero value of $A_{2n}$, followed by normalization via (29.6.5) and (29.6.6). Consequently, $\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i).

An alternative version of the Fourier series expansion (29.6.1) is given by

 29.6.8 $\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\mathop{% \mathrm{dn}\/}\nolimits\left(z,k\right)\left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{% \infty}C_{2p}\mathop{\cos\/}\nolimits\!\left(2p\phi\right)\right).$

Here $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ is as in §22.2, and

 29.6.9 $(\beta_{0}-H)C_{0}+\alpha_{0}C_{2}=0,$
 29.6.10 $\gamma_{p}C_{2p-2}+(\beta_{p}-H)C_{2p}+\alpha_{p}C_{2p+2}=0,$ $p\geq 1$,

with $\alpha_{p},\beta_{p}$, and $\gamma_{p}$ now defined by

 29.6.11 $\displaystyle\alpha_{p}$ $\displaystyle=\begin{cases}\nu(\nu+1)k^{2},&p=0,\\ \frac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},&p\geq 1,\end{cases}$ $\displaystyle\beta_{p}$ $\displaystyle=4p^{2}(2-k^{2}),$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2},$

and

 29.6.12 $\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty% }C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2}=1,$
 29.6.13 $\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p}>0,$ Symbols: $p$: nonnegative integer and $C_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E13 Encodings: TeX, pMML, png
 29.6.14 $\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+1$, or $\nu=2n+1$ and $m>n$,
 29.6.15 $\tfrac{1}{2}A_{0}C_{0}+\sum_{p=1}^{\infty}A_{2p}C_{2p}=\frac{4}{\pi}\int_{0}^{% \!\mathop{K\/}\nolimits\!}\left(\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!% \left(x,k^{2}\right)\right)^{2}dx.$

## §29.6(ii) Function $\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$

 29.6.16 $\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\sum_{p=0}^% {\infty}A_{2p+1}\mathop{\cos\/}\nolimits\!\left((2p+1)\phi\right).$

Here

 29.6.17 $H=2\!\mathop{a^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2},$ Defines: $H$ (locally) Symbols: $\mathop{a^{n}_{\nu}\/}\nolimits\!\left(k^{2}\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.6.E17 Encodings: TeX, pMML, png
 29.6.18 $(\beta_{0}-H)A_{1}+\alpha_{0}A_{3}=0,$
 29.6.19 $\gamma_{p}A_{2p-1}+(\beta_{p}-H)A_{2p+1}+\alpha_{p}A_{2p+3}=0,$ $p\geq 1$,

with $\alpha_{p}$, $\beta_{p}$, and $\gamma_{p}$ as in (29.3.13) and (29.3.14), and

 29.6.20 $\sum_{p=0}^{\infty}A_{2p+1}^{2}=1,$ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Referenced by: §29.6(ii) Permalink: http://dlmf.nist.gov/29.6.E20 Encodings: TeX, pMML, png
 29.6.21 $\sum_{p=0}^{\infty}A_{2p+1}>0,$ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Permalink: http://dlmf.nist.gov/29.6.E21 Encodings: TeX, pMML, png
 29.6.22 $\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+1$, or $\nu=2n+1$ and $m>n$.

Also,

 29.6.23 $\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\mathop{% \mathrm{dn}\/}\nolimits\left(z,k\right)\sum_{p=0}^{\infty}C_{2p+1}\mathop{\cos% \/}\nolimits\!\left((2p+1)\phi\right),$

where

 29.6.24 $(\beta_{0}-H)C_{1}+\alpha_{0}C_{3}=0,$
 29.6.25 $\gamma_{p}C_{2p-1}+(\beta_{p}-H)C_{2p+1}+\alpha_{p}C_{2p+3}=0,$ $p\geq 1$,

with

 29.6.26 $\displaystyle\alpha_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$ $\displaystyle\beta_{p}$ $\displaystyle=\begin{cases}2-k^{2}+\frac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},$

and

 29.6.27 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{\infty}C_{2p+1}^{2}-{\tfrac{1}{2}k% ^{2}\left(\tfrac{1}{2}C_{1}^{2}+\sum_{p=0}^{\infty}C_{2p+1}C_{2p+3}\right)=1},$
 29.6.28 $\sum_{p=0}^{\infty}C_{2p+1}>0,$ Symbols: $p$: nonnegative integer and $C_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E28 Encodings: TeX, pMML, png
 29.6.29 $\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+2$, or $\nu=2n+2$ and $m>n$,
 29.6.30 $\sum_{p=0}^{\infty}A_{2p+1}C_{2p+1}=\frac{4}{\pi}\int_{0}^{\!\mathop{K\/}% \nolimits\!}\left(\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(x,k^{2}% \right)\right)^{2}dx.$

## §29.6(iii) Function $\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$

 29.6.31 $\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\sum_{p=0}^% {\infty}B_{2p+1}\mathop{\sin\/}\nolimits\!\left((2p+1)\phi\right).$

Here

 29.6.32 $H=2\!\mathop{b^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2},$ Defines: $H$ (locally) Symbols: $\mathop{b^{n}_{\nu}\/}\nolimits\!\left(k^{2}\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.6.E32 Encodings: TeX, pMML, png
 29.6.33 $(\beta_{0}-H)B_{1}+\alpha_{0}B_{3}=0,$
 29.6.34 $\gamma_{p}B_{2p-1}+(\beta_{p}-H)B_{2p+1}+\alpha_{p}B_{2p+3}=0,$ $p\geq 1$,

with $\alpha_{p}$, $\beta_{p}$, and $\gamma_{p}$ as in (29.3.15), (29.3.16), and

 29.6.35 $\sum_{p=0}^{\infty}B_{2p+1}^{2}=1,$ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Referenced by: §29.6(iii) Permalink: http://dlmf.nist.gov/29.6.E35 Encodings: TeX, pMML, png
 29.6.36 $\sum_{p=0}^{\infty}(2p+1)B_{2p+1}>0,$ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E36 Encodings: TeX, pMML, png
 29.6.37 $\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+1$, or $\nu=2n+1$ and $m>n$.

Also,

 29.6.38 $\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\mathop{% \mathrm{dn}\/}\nolimits\left(z,k\right)\sum_{p=0}^{\infty}D_{2p+1}\mathop{\sin% \/}\nolimits\!\left((2p+1)\phi\right),$

where

 29.6.39 $(\beta_{0}-H)D_{1}+\alpha_{0}D_{3}=0,$
 29.6.40 $\gamma_{p}D_{2p-1}+(\beta_{p}-H)D_{2p+1}+\alpha_{p}D_{2p+3}=0,$ $p\geq 1$,

with

 29.6.41 $\displaystyle\alpha_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$ $\displaystyle\beta_{p}$ $\displaystyle=\begin{cases}2-k^{2}-\frac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2},$

and

 29.6.42 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{\infty}D_{2p+1}^{2}+{\tfrac{1}{2}k% ^{2}\left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{\infty}D_{2p+1}D_{2p+3}\right)=1},$
 29.6.43 $\sum_{p=0}^{\infty}(2p+1)D_{2p+1}>0,$ Symbols: $p$: nonnegative integer and $D_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E43 Encodings: TeX, pMML, png
 29.6.44 $\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+2$, or $\nu=2n+2$ and $m>n$,
 29.6.45 $\sum_{p=0}^{\infty}B_{2p+1}D_{2p+1}=\frac{4}{\pi}\int_{0}^{\!\mathop{K\/}% \nolimits\!}\left(\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(x,k^{2}% \right)\right)^{2}dx.$

## §29.6(iv) Function $\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$

 29.6.46 $\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\sum_{p=1}^% {\infty}B_{2p}\mathop{\sin\/}\nolimits\!\left(2p\phi\right).$

Here

 29.6.47 $H=2\!\mathop{b^{2m+2}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2},$ Defines: $H$ (locally) Symbols: $\mathop{b^{n}_{\nu}\/}\nolimits\!\left(k^{2}\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.6.E47 Encodings: TeX, pMML, png
 29.6.48 $(\beta_{0}-H)B_{2}+\alpha_{0}B_{4}=0,$
 29.6.49 $\gamma_{p}B_{2p}+(\beta_{p}-H)B_{2p+2}+\alpha_{p}B_{2p+4}=0,$ $p\geq 1$,

with $\alpha_{p}$, $\beta_{p}$, and $\gamma_{p}$ as in (29.3.17), and

 29.6.50 $\sum_{p=1}^{\infty}B_{2p}^{2}=1,$ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Referenced by: §29.6(iv) Permalink: http://dlmf.nist.gov/29.6.E50 Encodings: TeX, pMML, png
 29.6.51 $\sum_{p=0}^{\infty}(2p+2)B_{2p+2}>0,$ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E51 Encodings: TeX, pMML, png
 29.6.52 $\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+2$, or $\nu=2n+2$ and $m>n$.

Also,

 29.6.53 $\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)=\mathop{% \mathrm{dn}\/}\nolimits\left(z,k\right)\sum_{p=1}^{\infty}D_{2p}\mathop{\sin\/% }\nolimits\!\left(2p\phi\right),$

where

 29.6.54 $(\beta_{0}-H)D_{2}+\alpha_{0}D_{4}=0,$
 29.6.55 $\gamma_{p}D_{2p}+(\beta_{p}-H)D_{2p+2}+\alpha_{p}D_{2p+4}=0,$ $p\geq 1$,

with

 29.6.56 $\displaystyle\alpha_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$ $\displaystyle\beta_{p}$ $\displaystyle=(2p+2)^{2}(2-k^{2}),$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},$

and

 29.6.57 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2% }\sum_{p=1}^{\infty}D_{2p}D_{2p+2}=1,$
 29.6.58 $\sum_{p=0}^{\infty}(2p+2)D_{2p+2}>0,$ Symbols: $p$: nonnegative integer and $D_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E58 Encodings: TeX, pMML, png
 29.6.59 $\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}}=\frac{k^{2}}{(1+k^{\prime})^{2}},$ $\nu\neq 2n+3$, or $\nu=2n+3$ and $m>n$,
 29.6.60 $\sum_{p=1}^{\infty}B_{2p}D_{2p}=\frac{4}{\pi}\int_{0}^{\!\mathop{K\/}\nolimits% \!}\left(\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(x,k^{2}\right)% \right)^{2}dx.$