# §29.3 Definitions and Basic Properties

## §29.3(i) Eigenvalues

For each pair of values of and there are four infinite unbounded sets of real eigenvalues for which equation (29.2.1) has even or odd solutions with periods or . They are denoted by , , , , where ; see Table 29.3.1.

## §29.3(ii) Distribution

The eigenvalues interlace according to

The eigenvalues coalesce according to

If is distinct from , then

If is a nonnegative integer, then

For the special case see Erdélyi et al. (1955, §15.5.2).

## §29.3(iii) Continued Fractions

The quantity

satisfies the continued-fraction equation

29.3.10

where 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 , and if , then the last denominator is . If is a nonnegative integer and , 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 . If is a nonnegative integer and , then (29.3.10) has only the solutions (29.3.9) with .

The quantity satisfies equation (29.3.10) with

The quantity satisfies equation (29.3.10) with

The quantity satisfies equation (29.3.10) with

## §29.3(iv) Lamé Functions

The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by , , , . They are called Lamé functions with real periods and of order , or more simply, Lamé functions. See Table 29.3.2. In this table the nonnegative integer corresponds to the number of zeros of each Lamé function in , whereas the superscripts , , or correspond to the number of zeros in .

Table 29.3.2: Lamé functions.
boundary conditions
 eigenvalue
 eigenfunction
 parity of
 parity of
 period of
even even
odd even
even odd
odd odd

## §29.3(v) Normalization

To complete the definitions, is positive and is negative.

## §29.3(vii) Power Series

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