Lamé polynomials are orthogonal in two ways. First, the orthogonality relations (29.3.19) apply; see §29.12(i). Secondly, the system of functions
is orthogonal and complete with respect to the inner product
Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2):
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
with given by (29.14.3).