§29.14 Orthogonality

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

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 $f_{n}^{m}(s,t)=\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right),$ $n=0,1,2,\dots$ , $m=0,1,\dots,n$ ,

Symbols:: $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : real parameter and $f_{n}^{m}(s,t)$ : system
Referenced by:: §29.14
Permalink:: http://dlmf.nist.gov/29.14.E1
Encodings:: TeX, pMML, png

is orthogonal and complete with respect to the inner product

29.14.2 $\langle g,h\rangle=\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\!\!\int _{0}^{{\!\mathop{{K^{{\prime}}}\/}\nolimits\!}}w(s,t)g(s,t)h(s,t)dtds,$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

where

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

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $k$ : real parameter and $w(s,t)$ : function
Referenced by:: §29.14
Permalink:: http://dlmf.nist.gov/29.14.E3
Encodings:: TeX, pMML, png

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

29.14.4 $\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right),$

Symbols:: $\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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.E4
Encodings:: TeX, pMML, png

29.14.5 $\mathop{\mathit{cE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{cE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right),$

Symbols:: $\mathop{\mathit{cE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer, $n$ : nonnegative integer and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.14.E5
Encodings:: TeX, pMML, png

29.14.6 $\mathop{\mathit{dE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{dE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right),$

Symbols:: $\mathop{\mathit{dE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer, $n$ : nonnegative integer and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.14.E6
Encodings:: TeX, pMML, png

29.14.7 $\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right),$

Symbols:: $\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer, $n$ : nonnegative integer and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.14.E7
Encodings:: TeX, pMML, png

29.14.8 $\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right),$

Symbols:: $\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer, $n$ : nonnegative integer and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.14.E8
Encodings:: TeX, pMML, png

29.14.9 $\mathop{\mathit{cdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{cdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right),$

Symbols:: $\mathop{\mathit{cdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer, $n$ : nonnegative integer and $k$ : real parameter
Permalink:: http://dlmf.nist.gov/29.14.E9
Encodings:: TeX, pMML, png

29.14.10 $\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(s,k^{2}\right)\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!+it,k^{2}\right).$

Symbols:: $\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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.E10
Encodings:: TeX, pMML, png

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 $\langle g,h\rangle=\int _{0}^{{4\!\mathop{K\/}\nolimits\!}}\!\!\int _{0}^{{2\!\mathop{{K^{{\prime}}}\/}\nolimits\!}}w(s,t)g(s,t)h(s,t)dtds,$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

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