# §29.1 Special Notation

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

$m,n,p$ nonnegative integers. real variable. complex variable. real parameters, $0, $\nu\geq-\frac{1}{2}$. $\sqrt{1-k^{2}}$, $0. complete elliptic integrals of the first kind with moduli $k,k^{\prime}$, respectively (see §19.2(ii)).

All derivatives are denoted by differentials, not by primes.

The main functions treated in this chapter are the eigenvalues $\mathop{a^{2m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $\mathop{a^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $\mathop{b^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $\mathop{b^{2m+2}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, the Lamé functions $\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, and the Lamé polynomials $\mathop{\mathit{uE}^{m}_{2n}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{sE}^{m}_{2n+1}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{cE}^{m}_{2n+1}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{dE}^{m}_{2n+1}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{scE}^{m}_{2n+2}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{sdE}^{m}_{2n+2}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{cdE}^{m}_{2n+2}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{scdE}^{m}_{2n+3}\/}\nolimits\!\left(z,k^{2}\right)$. 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 $\mathop{a^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)$ with $\mathop{b^{2m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)$. The relation to the Lamé functions $L^{(m)}_{c\nu}$, $L^{(m)}_{s\nu}$of Jansen (1977) is given by

 $\displaystyle\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=(-1)^{m}L_{c\nu}^{(2m)}(\psi,{k^{\prime}}^{2}),$ $\displaystyle\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=(-1)^{m}L_{s\nu}^{(2m+1)}(\psi,{k^{\prime}}^{2}),$ $\displaystyle\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=(-1)^{m}L_{c\nu}^{(2m+1)}(\psi,{k^{\prime}}^{2}),$ $\displaystyle\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=(-1)^{m}L_{s\nu}^{(2m+2)}(\psi,{k^{\prime}}^{2}),$

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

 $\displaystyle\mathop{\mathit{Ec}^{2m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=c_{\nu}^{2m}(k^{2}){\rm Ec}_{\nu}^{2m}(z,k^{2}),$ $\displaystyle\mathop{\mathit{Ec}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=c_{\nu}^{2m+1}(k^{2}){\rm Es}_{\nu}^{2m+1}(z,k^{2}),$ $\displaystyle\mathop{\mathit{Es}^{2m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=s_{\nu}^{2m+1}(k^{2}){\rm Ec}_{\nu}^{2m+1}(z,k^{2}),$ $\displaystyle\mathop{\mathit{Es}^{2m+2}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=s_{\nu}^{2m+2}(k^{2}){\rm Es}_{\nu}^{2m+2}(z,k^{2}),$

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

 $\displaystyle(c_{\nu}^{m}(k^{2}))^{2}$ $\displaystyle=\frac{4}{\pi}\int_{0}^{\!\mathop{K\/}\nolimits\!}\left(\mathop{% \mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(x,k^{2}\right)\right)^{2}dx,$ $\displaystyle(s_{\nu}^{m}(k^{2}))^{2}$ $\displaystyle=\frac{4}{\pi}\int_{0}^{\!\mathop{K\/}\nolimits\!}\left(\mathop{% \mathit{Es}^{m}_{\nu}\/}\nolimits\!\left(x,k^{2}\right)\right)^{2}dx.$