29 Lamé Functions Lamé Polynomials 29.13 Graphics 29.15 Fourier Series and Chebyshev Series

§29.14 Orthogonality

ⓘ

Keywords:: Lamé polynomials, orthogonality
Notes:: See Arscott (1964b, §9.4) and Erdélyi et al. (1955, §15.7).
Permalink:: http://dlmf.nist.gov/29.14
See also:: Annotations for Ch.29

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		$\langle g,h\rangle=\int_{0}^{K}\!\!\int_{0}^{{K^{\prime}}}w(s,t)g(s,t)h(s,t)\,% \mathrm{d}t\,\mathrm{d}s,$
ⓘ Symbols: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : 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		$w(s,t)={\operatorname{sn}}^{2}\left(K+\mathrm{i}t,k\right)-{\operatorname{sn}}% ^{2}\left(s,k\right).$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : 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		$\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
ⓘ Symbols: $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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.E4 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.5		$\mathit{cE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{cE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
ⓘ Symbols: $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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 Permalink: http://dlmf.nist.gov/29.14.E5 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.6		$\mathit{dE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{dE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
ⓘ Symbols: $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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 Permalink: http://dlmf.nist.gov/29.14.E6 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.7		$\mathit{scE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{scE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$
ⓘ Symbols: $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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 Permalink: http://dlmf.nist.gov/29.14.E7 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.8		$\mathit{sdE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{sdE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$
ⓘ Symbols: $\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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 Permalink: http://dlmf.nist.gov/29.14.E8 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.9		$\mathit{cdE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{cdE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$
ⓘ Symbols: $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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 Permalink: http://dlmf.nist.gov/29.14.E9 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29
29.14.10		$\mathit{scdE}^{m}_{2n+3}\left(s,k^{2}\right)\mathit{scdE}^{m}_{2n+3}\left(K+% \mathrm{i}t,k^{2}\right).$
ⓘ Symbols: $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : 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.E10 Encodings: TeX, pMML, png See also: Annotations for §29.14 and Ch.29

In each system $n$ ranges over all nonnegative integers and $m=0,1,\dots,n$ . 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		$\langle g,h\rangle=\int_{0}^{4K}\!\!\int_{0}^{2{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \,\mathrm{d}t\,\mathrm{d}s,$
ⓘ Symbols: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\,\mathrm{d}\NVar{x}$ : 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).

29.13 Graphics 29.15 Fourier Series and Chebyshev Series

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