Lamé polynomials are orthogonal in two ways. First, the orthogonality relations (29.3.19) apply; see §29.12(i). Secondly, the system of functions
29.14.1 | |||
, , | |||
is orthogonal and complete with respect to the inner product
29.14.2 | |||
where
29.14.3 | |||
Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2):
29.14.4 | |||
29.14.5 | |||
29.14.6 | |||
29.14.7 | |||
29.14.8 | |||
29.14.9 | |||
29.14.10 | |||
In each system ranges over all nonnegative integers and . When combined, all eight systems (29.14.1) and (29.14.4)–(29.14.10) form an orthogonal and complete system with respect to the inner product
29.14.11 | |||
with given by (29.14.3).