§29.3 Definitions and Basic Properties

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 $2K$ or $4K$. They are denoted by $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, where $m=0,1,2,\mathrm{\dots }$; see Table 29.3.1.

The eigenvalues interlace according to

The eigenvalues coalesce according to

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

If $\nu$ is a nonnegative integer, then

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

The quantity

satisfies the continued-fraction equation

where $p$ is any nonnegative integer, and

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 $2p\le \nu$, 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 $2m\le \nu$. 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 $H=2a_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with

The quantity $H=2b_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with

The quantity $H=2b_{\nu }^{2m+2}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with

The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$. 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 $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$.

For $\mathrm{dn}(z,k)$ see §22.2.

To complete the definitions, ${\mathrm{𝐸𝑐}}_{\nu }^m(K,k^2)$ is positive and ${d{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)/dz|}_{z=K}$ is negative.

For $m\ne p$,

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

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