§29.1 Special Notation

(For other notation see Notation for the Special Functions.)

All derivatives are denoted by differentials, not by primes.

The main functions treated in this chapter are the eigenvalues $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$, and the Lamé polynomials ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$. The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). The normalization is that of Jansen (1977, §3.1).

Other notations that have been used are as follows: Ince (1940a) interchanges $a_{\nu }^{2m+1}\left(k^2\right)$ with $b_{\nu }^{2m+1}\left(k^2\right)$. The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by

where $\psi =\mathrm{am}(z,k)$; see §22.16(i). The relation to the Lamé functions ${\mathrm{Ec}}_{\nu }^m$, ${\mathrm{Es}}_{\nu }^m$ of Ince (1940b) is given by

where the positive factors $c_{\nu }^m(k^2)$ and $s_{\nu }^m(k^2)$ are determined by