# §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 $f_{n}^{m}(s,t)=\mathit{uE}^{m}_{2n}\left(s,k^{2}\right)\mathit{uE}^{m}_{2n}% \left(K+\mathrm{i}t,k^{2}\right),$ $n=0,1,2,\dots$, $m=0,1,\dots,n$, ⓘ Defines: $f_{n}^{m}(s,t)$: system (locally) Symbols: $\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial, $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter Referenced by: §29.14 Permalink: http://dlmf.nist.gov/29.14.E1 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29

is orthogonal and complete with respect to the inner product

 29.14.2 $\langle g,h\rangle=\int_{0}^{K}\!\!\int_{0}^{{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \mathrm{d}t\mathrm{d}s,$

where

 29.14.3 $w(s,t)={\operatorname{sn}}^{2}\left(K+\mathrm{i}t,k\right)-{\operatorname{sn}}% ^{2}\left(s,k\right).$

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

 29.14.4 $\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$ 29.14.5 $\mathit{cE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{cE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$ 29.14.6 $\mathit{dE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{dE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$ 29.14.7 $\mathit{scE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{scE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$ 29.14.8 $\mathit{sdE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{sdE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$ 29.14.9 $\mathit{cdE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{cdE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$ 29.14.10 $\mathit{scdE}^{m}_{2n+3}\left(s,k^{2}\right)\mathit{scdE}^{m}_{2n+3}\left(K+% \mathrm{i}t,k^{2}\right).$

In each system $n$ ranges over all nonnegative integers and $m=0,1,\dots,n$. 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 $\langle g,h\rangle=\int_{0}^{4K}\!\!\int_{0}^{2{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \mathrm{d}t\mathrm{d}s,$

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