§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		$$f_n^m(s,t)={\mathit{uE}}_{2n}^m(s,k^2){\mathit{uE}}_{2n}^m(K+\mathrm{i}t,k^2),$$
$n=0,1,2,\mathrm{\dots }$, $m=0,1,\mathrm{\dots },n$,
ⓘ Defines: $f_n^m(s,t)$: system (locally) Symbols: ${\mathit{uE}}_{2n}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: 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		$$⟨g,h⟩={\int }_0^K{\int }_0^{K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$
ⓘ Symbols: $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $k$: real parameter and $w(s,t)$: function Referenced by: §29.14 Permalink: http://dlmf.nist.gov/29.14.E2 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29

where

29.14.3		$$w(s,t)={\mathrm{sn}}^2(K+\mathrm{i}t,k)-{\mathrm{sn}}^2(s,k).$$
ⓘ Symbols: $\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: real parameter and $w(s,t)$: function Referenced by: §29.14 Permalink: http://dlmf.nist.gov/29.14.E3 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29

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

29.14.4		$${\mathit{sE}}_{2n+1}^m(s,k^2){\mathit{sE}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$
ⓘ Symbols: ${\mathit{sE}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: 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.E4 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.5		$${\mathit{cE}}_{2n+1}^m(s,k^2){\mathit{cE}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$
ⓘ Symbols: ${\mathit{cE}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter Permalink: http://dlmf.nist.gov/29.14.E5 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.6		$${\mathit{dE}}_{2n+1}^m(s,k^2){\mathit{dE}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$
ⓘ Symbols: ${\mathit{dE}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter Permalink: http://dlmf.nist.gov/29.14.E6 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.7		$${\mathit{scE}}_{2n+2}^m(s,k^2){\mathit{scE}}_{2n+2}^m(K+\mathrm{i}t,k^2),$$
ⓘ Symbols: ${\mathit{scE}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter Permalink: http://dlmf.nist.gov/29.14.E7 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.8		$${\mathit{sdE}}_{2n+2}^m(s,k^2){\mathit{sdE}}_{2n+2}^m(K+\mathrm{i}t,k^2),$$
ⓘ Symbols: ${\mathit{sdE}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter Permalink: http://dlmf.nist.gov/29.14.E8 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.9		$${\mathit{cdE}}_{2n+2}^m(s,k^2){\mathit{cdE}}_{2n+2}^m(K+\mathrm{i}t,k^2),$$
ⓘ Symbols: ${\mathit{cdE}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter Permalink: http://dlmf.nist.gov/29.14.E9 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.10		$${\mathit{scdE}}_{2n+3}^m(s,k^2){\mathit{scdE}}_{2n+3}^m(K+\mathrm{i}t,k^2).$$
ⓘ Symbols: ${\mathit{scdE}}_{2n+3}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: 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.E10 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29

In each system $n$ ranges over all nonnegative integers and $m=0,1,\mathrm{\dots },n$. 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		$$⟨g,h⟩={\int }_0^{4K}{\int }_0^{2K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$
ⓘ Symbols: $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $k$: real parameter and $w(s,t)$: function Permalink: http://dlmf.nist.gov/29.14.E11 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29

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