# §29.12 Definitions

## §29.12(i) Elliptic-Function Form

Throughout §§29.1229.16 the order in the differential equation (29.2.1) is assumed to be a nonnegative integer.

The Lamé functions , , and , , are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows:

29.12.1
29.12.8

where , . These functions are polynomials in , , and . In consequence they are doubly-periodic meromorphic functions of .

The superscript on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of -zeros of each Lamé polynomial in the interval , while is the number of -zeros in the open line segment from to .

The prefixes , , , , , , , 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 and modulus of the Jacobian elliptic functions have been suppressed, and denotes a polynomial of degree in (different for each type). For the determination of the coefficients of the ’s see §29.15(ii).

Table 29.12.1: Lamé polynomials.
 eigenvalue
 eigenfunction
 polynomial form
 real period
 imag. period
 parity of
 parity of
 parity of
even even even
odd even even
even odd even
even even odd
odd odd even
odd even odd
even odd odd
odd odd odd

## §29.12(ii) Algebraic Form

With the substitution every Lamé polynomial in Table 29.12.1 can be written in the form

where , , are either 0 or . The polynomial is of degree and has zeros (all simple) in and zeros (all simple) in . The functions (29.12.9) satisfy (29.2.2).

## §29.12(iii) Zeros

Let denote the zeros of the polynomial in (29.12.9) arranged according to

Then the function

defined for with

attains its absolute maximum iff , . Moreover,

This result admits the following electrostatic interpretation: Given three point masses fixed at , , and with positive charges , , and , respectively, and movable point masses at arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when for .