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

§29.14 Orthogonality

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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 $f_{n}^{m}(s,t)=\mathit{uE}^{m}_{2n}\left(s,k^{2}\right)\mathit{uE}^{m}_{2n}% \left(K+\mathrm{i}t,k^{2}\right),$ $n=0,1,2,\dots$ , $m=0,1,\dots,n$ ,

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

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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 , $\int$ : integral, : real parameter and : 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 $w(s,t)={\operatorname{sn}}^{2}\left(K+\mathrm{i}t,k\right)-{\operatorname{sn}}% ^{2}\left(s,k\right).$

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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, : real parameter and : 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 $\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$

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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, : nonnegative integer, : nonnegative integer and : 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 $\mathit{cE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{cE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$

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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, : nonnegative integer, : nonnegative integer and : 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 $\mathit{dE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{dE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$

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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, : nonnegative integer, : nonnegative integer and : 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 $\mathit{scE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{scE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$

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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, : nonnegative integer, : nonnegative integer and : 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 $\mathit{sdE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{sdE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$

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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, : nonnegative integer, : nonnegative integer and : 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 $\mathit{cdE}^{m}_{2n+2}\left(s,k^{2}\right)\mathit{cdE}^{m}_{2n+2}\left(K+% \mathrm{i}t,k^{2}\right),$

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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, : nonnegative integer, : nonnegative integer and : 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 $\mathit{scdE}^{m}_{2n+3}\left(s,k^{2}\right)\mathit{scdE}^{m}_{2n+3}\left(K+% \mathrm{i}t,k^{2}\right).$

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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, : nonnegative integer, : nonnegative integer and : 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 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 $\langle g,h\rangle=\int_{0}^{4K}\!\!\int_{0}^{2{K^{\prime}}}w(s,t)g(s,t)h(s,t)% \,\mathrm{d}t\,\mathrm{d}s,$

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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 , $\int$ : integral, : real parameter and : function
Permalink:: http://dlmf.nist.gov/29.14.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §29.14 and Ch.29

with given by (29.14.3).

29.13 Graphics 29.15 Fourier Series and Chebyshev Series

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