§29.12 Definitions

Throughout §§29.12–29.16 the order $\nu$ in the differential equation (29.2.1) is assumed to be a nonnegative integer.

The Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, $m=0,1,\mathrm{\dots },\nu$, and ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, $m=1,2,\mathrm{\dots },\nu$, are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows:

where $n=0,1,2,\mathrm{\dots }$, $m=0,1,2,\mathrm{\dots },n$. These functions are polynomials in $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, and $\mathrm{dn}(z,k)$. In consequence they are doubly-periodic meromorphic functions of $z$.

The superscript $m$ on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of $z$-zeros of each Lamé polynomial in the interval $(0,K)$, while $n-m$ is the number of $z$-zeros in the open line segment from $K$ to $K+\mathrm{i}{K}^{\prime }$.

The prefixes $u$, $s$, $c$, $d$, $\mathrm{𝑠𝑐}$, $\mathrm{𝑠𝑑}$, $\mathrm{𝑐𝑑}$, $\mathrm{𝑠𝑐𝑑}$ indicate the type of the polynomial form of the Lamé polynomial; compare the 3rd and 4th columns in Table 29.12.1. In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). For the determination of the coefficients of the $P$’s see §29.15(ii).

With the substitution $\xi ={\mathrm{sn}}^2(z,k)$ every Lamé polynomial in Table 29.12.1 can be written in the form

where $\rho$, $\sigma$, $\tau$ are either $0$ or $\frac{1}{2}$. The polynomial $P(\xi )$ is of degree $n$ and has $m$ zeros (all simple) in $(0,1)$ and $n-m$ zeros (all simple) in $(1,k^{-2})$. The functions (29.12.9) satisfy (29.2.2).

Let ${\xi }_1,{\xi }_2,\mathrm{\dots },{\xi }_n$ denote the zeros of the polynomial $P$ in (29.12.9) arranged according to

Then the function

defined for $(t_1,t_2,\mathrm{\dots },t_n)$ with

attains its absolute maximum iff $t_j={\xi }_j$, $j=1,2,\mathrm{\dots },n$. Moreover,

This result admits the following electrostatic interpretation: Given three point masses fixed at $t=0$, $t=1$, and $t=k^{-2}$ with positive charges $\rho +\frac{1}{4}$, $\sigma +\frac{1}{4}$, and $\tau +\frac{1}{4}$, respectively, and $n$ movable point masses at $t_1,t_2,\mathrm{\dots },t_n$ arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when $t_j={\xi }_j$ for $j=1,2,\mathrm{\dots },n$.