When , , the Fourier series (29.6.1) terminates:
A convenient way of constructing the coefficients, together with the eigenvalues, is as follows. Equations (29.6.4), with , (29.6.3), and can be cast as an algebraic eigenvalue problem in the following way. Let
be the tridiagonal matrix with , , as in (29.3.11), (29.3.12). Let the eigenvalues of be with
and also let
be the eigenvector corresponding to and normalized so that
The Chebyshev polynomial of the first kind (§18.3) satisfies . Since (29.2.5) implies that , (29.15.1) can be rewritten in the form
This determines the polynomial of degree for which ; compare Table 29.12.1. The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an tridiagonal matrix; see Arscott and Khabaza (1962).
Using also , with denoting the Chebyshev polynomial of the second kind (§18.3), we obtain
For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).