When , , the Fourier series (29.6.1) terminates:
29.15.1 | |||
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
29.15.2 | |||
be the tridiagonal matrix with , , as in (29.3.11), (29.3.12). Let the eigenvalues of be with
29.15.3 | |||
and also let
29.15.4 | |||
be the eigenvector corresponding to and normalized so that
29.15.5 | |||
and
29.15.6 | |||
Then
29.15.7 | |||
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
29.15.43 | |||
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
29.15.44 | ||||
29.15.45 | ||||
29.15.46 | ||||
29.15.47 | ||||
29.15.48 | ||||
29.15.49 | ||||
29.15.50 | ||||
For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).