§29.3 Definitions and Basic Properties

§29.3(i) Eigenvalues

For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2\!\mathop{K\/}\nolimits\!$ or $4\!\mathop{K\/}\nolimits\!$. They are denoted by $\mathop{a^{2m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $\mathop{a^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $\mathop{b^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $\mathop{b^{2m+2}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, where $m=0,1,2,\ldots$; see Table 29.3.1.

§29.3(ii) Distribution

The eigenvalues interlace according to

 29.3.1 $\displaystyle\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ $\displaystyle<\mathop{a^{m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right),$ 29.3.2 $\displaystyle\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ $\displaystyle<\mathop{b^{m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right),$ 29.3.3 $\displaystyle\mathop{b^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ $\displaystyle<\mathop{b^{m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right),$ 29.3.4 $\displaystyle\mathop{b^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ $\displaystyle<\mathop{a^{m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right).$

The eigenvalues coalesce according to

 29.3.5 $\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)=\mathop{b^{m}_{\nu}\/}% \nolimits\!\left(k^{2}\right),$ $\nu=0,1,\dots,m-1$.

If $\nu$ is distinct from $0,1,\dots,m-1$, then

 29.3.6 $\left(\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\mathop{b^{m}_{\nu}% \/}\nolimits\!\left(k^{2}\right)\right)\nu(\nu-1)\cdots(\nu-m+1)>0.$

If $\nu$ is a nonnegative integer, then

 29.3.7 $\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)+\mathop{a^{\nu-m}_{\nu}\/}% \nolimits\!\left(1-k^{2}\right)=\nu(\nu+1),$ $m=0,1,\dots,\nu$,
 29.3.8 ${\mathop{b^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)+\mathop{b^{\nu-m+1}_{\nu}% \/}\nolimits\!\left(1-k^{2}\right)=\nu(\nu+1)},$ $m=1,2,\dots,\nu$.

For the special case $k=k^{\prime}=\ifrac{1}{\sqrt{2}}$ see Erdélyi et al. (1955, §15.5.2).

§29.3(iii) Continued Fractions

The quantity

 29.3.9 $H=2\!\mathop{a^{2m}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ Defines: $H$ (locally) Symbols: $\mathop{a^{\NVar{n}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{k^{2}}\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.3(iii) Permalink: http://dlmf.nist.gov/29.3.E9 Encodings: TeX, pMML, png See also: Annotations for 29.3(iii)

satisfies the continued-fraction equation

 29.3.10 $\beta_{p}-H-\cfrac{\alpha_{p-1}\gamma_{p}}{\beta_{p-1}-H-\cfrac{\alpha_{p-2}% \gamma_{p-1}}{\beta_{p-2}-H-}}\dots=\cfrac{\alpha_{p}\gamma_{p+1}}{\beta_{p+1}% -H-\cfrac{\alpha_{p+1}\gamma_{p+2}}{\beta_{p+2}-H-}}\cdots,$ Symbols: $p$: nonnegative integer, $H$, $\alpha_{p}$, $\beta_{p}$ and $\gamma_{p}$ Referenced by: §29.20(i), §29.3(iii), §29.3(iii), §29.3(iii), §29.3(iii) Permalink: http://dlmf.nist.gov/29.3.E10 Encodings: TeX, pMML, png See also: Annotations for 29.3(iii)

where $p$ is any nonnegative integer, and

 29.3.11 $\alpha_{p}=\begin{cases}(\nu-1)(\nu+2)k^{2},&p=0,\\ \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},&p\geq 1,\end{cases}$ Defines: $\alpha_{p}$ (locally) Symbols: $p$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.15(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.3.E11 Encodings: TeX, pMML, png See also: Annotations for 29.3(iii)
 29.3.12 $\displaystyle\beta_{p}$ $\displaystyle=4p^{2}(2-k^{2}),$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p+2)(\nu+2p-1)k^{2}.$ Defines: $\beta_{p}$ (locally) and $\gamma_{p}$ (locally) Symbols: $p$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.15(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.3.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 29.3(iii)

The continued fraction following the second negative sign on the left-hand side of (29.3.10) is finite: it equals 0 if $p=0$, and if $p>0$, then the last denominator is $\beta_{0}-H$. If $\nu$ is a nonnegative integer and $2p\leq\nu$, then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with $2m\leq\nu$. If $\nu$ is a nonnegative integer and $2p>\nu$, then (29.3.10) has only the solutions (29.3.9) with $2m>\nu$.

The quantity $H=2\!\mathop{a^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

 29.3.13 $\beta_{p}=\begin{cases}2-k^{2}+\tfrac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $\beta_{p}$ Referenced by: §29.15(i), §29.6(ii) Permalink: http://dlmf.nist.gov/29.3.E13 Encodings: TeX, pMML, png See also: Annotations for 29.3(iii)
 29.3.14 $\displaystyle\alpha_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}.$ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\alpha_{p}$ and $\gamma_{p}$ Referenced by: §29.15(i), §29.6(ii) Permalink: http://dlmf.nist.gov/29.3.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 29.3(iii)

The quantity $H=2\!\mathop{b^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

 29.3.15 $\beta_{p}=\begin{cases}2-k^{2}-\tfrac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $\beta_{p}$ Referenced by: §29.15(i), §29.6(iii) Permalink: http://dlmf.nist.gov/29.3.E15 Encodings: TeX, pMML, png See also: Annotations for 29.3(iii)
 29.3.16 $\displaystyle\alpha_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}.$ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\alpha_{p}$ and $\gamma_{p}$ Referenced by: §29.15(i), §29.6(iii) Permalink: http://dlmf.nist.gov/29.3.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 29.3(iii)

The quantity $H=2\!\mathop{b^{2m+2}_{\nu}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

 29.3.17 $\displaystyle\alpha_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p-3)(\nu+2p+4)k^{2},$ $\displaystyle\beta_{p}$ $\displaystyle=(2p+2)^{2}(2-k^{2}),$ $\displaystyle\gamma_{p}$ $\displaystyle=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}.$ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\alpha_{p}$, $\beta_{p}$ and $\gamma_{p}$ Referenced by: §29.15(i), §29.6(iv) Permalink: http://dlmf.nist.gov/29.3.E17 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 29.3(iii)

§29.3(iv) Lamé Functions

The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by $\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$. They are called Lamé functions with real periods and of order $\nu$, or more simply, Lamé functions. See Table 29.3.2. In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,\!\mathop{K\/}\nolimits\!)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2\!\mathop{K\/}\nolimits\!)$.

§29.3(v) Normalization

 29.3.18 $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathrm{dn}\/}% \nolimits\left(x,k\right)\left(\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!% \left(x,k^{2}\right)\right)^{2}\mathrm{d}x$ $\displaystyle=\frac{1}{4}\pi,$ $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathrm{dn}\/}% \nolimits\left(x,k\right)\left(\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!% \left(x,k^{2}\right)\right)^{2}\mathrm{d}x$ $\displaystyle=\frac{1}{4}\pi,$ $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathrm{dn}\/}% \nolimits\left(x,k\right)\left(\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!% \left(x,k^{2}\right)\right)^{2}\mathrm{d}x$ $\displaystyle=\frac{1}{4}\pi,$ $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathrm{dn}\/}% \nolimits\left(x,k\right)\left(\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!% \left(x,k^{2}\right)\right)^{2}\mathrm{d}x$ $\displaystyle=\frac{1}{4}\pi.$

For $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ see §22.2.

To complete the definitions, $\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!,k^{% 2}\right)$ is positive and $\left.\ifrac{\mathrm{d}\mathop{\mathit{Es}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2% }\right)}{\mathrm{d}z}\right|_{z=\!\mathop{K\/}\nolimits\!}$ is negative.

§29.3(vi) Orthogonality

For $m\neq p$,

 29.3.19 $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathit{Ec}^{2m}_{\nu% }\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Ec}^{2p}_{\nu}\/}\nolimits% \!\left(x,k^{2}\right)\mathrm{d}x$ $\displaystyle=0,$ $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathit{Ec}^{2m+1}_{% \nu}\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Ec}^{2p+1}_{\nu}\/}% \nolimits\!\left(x,k^{2}\right)\mathrm{d}x$ $\displaystyle=0,$ $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathit{Es}^{2m+1}_{% \nu}\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Es}^{2p+1}_{\nu}\/}% \nolimits\!\left(x,k^{2}\right)\mathrm{d}x$ $\displaystyle=0,$ $\displaystyle\int_{0}^{\!\mathop{K\/}\nolimits\!}\mathop{\mathit{Es}^{2m+2}_{% \nu}\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Es}^{2p+2}_{\nu}\/}% \nolimits\!\left(x,k^{2}\right)\mathrm{d}x$ $\displaystyle=0.$

For the values of these integrals when $m=p$ see §29.6.

§29.3(vii) Power Series

For power-series expansions of the eigenvalues see Volkmer (2004b).