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
and also let
be the eigenvector corresponding to and normalized so that
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).