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29 Lamé FunctionsLamé Polynomials

§29.14 Orthogonality

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 fnm(s,t)=uE2nm(s,k2)uE2nm(K+it,k2),
n=0,1,2,, m=0,1,,n,

is orthogonal and complete with respect to the inner product

29.14.2 g,h=0K0Kw(s,t)g(s,t)h(s,t)dtds,

where

29.14.3 w(s,t)=sn2(K+it,k)-sn2(s,k).

Each of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2):

29.14.4 sE2n+1m(s,k2)sE2n+1m(K+it,k2),
29.14.5 cE2n+1m(s,k2)cE2n+1m(K+it,k2),
29.14.6 dE2n+1m(s,k2)dE2n+1m(K+it,k2),
29.14.7 scE2n+2m(s,k2)scE2n+2m(K+it,k2),
29.14.8 sdE2n+2m(s,k2)sdE2n+2m(K+it,k2),
29.14.9 cdE2n+2m(s,k2)cdE2n+2m(K+it,k2),
29.14.10 scdE2n+3m(s,k2)scdE2n+3m(K+it,k2).

In each system n ranges over all nonnegative integers and m=0,1,,n. 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 g,h=04K02Kw(s,t)g(s,t)h(s,t)dtds,

with w(s,t) given by (29.14.3).