§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 $f_{n}^{m}(s,t)=\mathop{\mathit{uE}^{m}_{2n}\/}\nolimits\!\left(s,k^{2}\right)% \mathop{\mathit{uE}^{m}_{2n}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+% \mathrm{i}t,k^{2}\right),$ $n=0,1,2,\dots$, $m=0,1,\dots,n$, Defines: $f_{n}^{m}(s,t)$: system (locally) Symbols: $\mathop{\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{k% ^{2}}\right)$: Lamé polynomial, $\mathop{K\/}\nolimits\!\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $m$: 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

is orthogonal and complete with respect to the inner product

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

where

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

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

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

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 $\langle g,h\rangle=\int_{0}^{4\!\mathop{K\/}\nolimits\!}\!\!\int_{0}^{2\!% \mathop{{K^{\prime}}\/}\nolimits\!}w(s,t)g(s,t)h(s,t)\mathrm{d}t\mathrm{d}s,$

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