29 Lamé Functions Lamé Polynomials 29.14 Orthogonality 29.16 Asymptotic Expansions

§29.15 Fourier Series and Chebyshev Series

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Keywords:: Fourier series, Lamé polynomials
Permalink:: http://dlmf.nist.gov/29.15
See also:: Annotations for Ch.29

§29.15(i) Fourier Coefficients
§29.15(ii) Chebyshev Series

§29.15(i) Fourier Coefficients

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Notes:: The method for constructing the coefficients follows from §29.6.
Referenced by:: §29.20(i), §29.20(ii), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.15.i
See also:: Annotations for §29.15 and Ch.29

Polynomial $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$

ⓘ

See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n$ , $m=0,1,\dots,n$ , the Fourier series (29.6.1) terminates:

29.15.1 $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p% }\cos\left(2p\phi\right).$

ⓘ

Symbols:: $\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $A_{2p}$ : coefficients
Referenced by:: §29.15(i), §29.15(ii)
Permalink:: http://dlmf.nist.gov/29.15.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

A convenient way of constructing the coefficients, together with the eigenvalues, is as follows. Equations (29.6.4), with $p=1,2,\dots,n$ , (29.6.3), and $A_{2n+2}=0$ can be cast as an algebraic eigenvalue problem in the following way. Let

29.15.2 $\mathbf{M}=\begin{bmatrix}\beta_{0}&\alpha_{0}&0&\cdots&0\\ \gamma_{1}&\beta_{1}&\alpha_{1}&\ddots&\vdots\\ 0&\ddots&\ddots&\ddots&0\\ \vdots&\ddots&\gamma_{n-1}&\beta_{n-1}&\alpha_{n-1}\\ 0&\cdots&0&\gamma_{n}&\beta_{n}\end{bmatrix}$

ⓘ

Symbols:: : nonnegative integer, $\alpha_{p}$ , $\beta_{p}$ and $\gamma_{p}$
Referenced by:: §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

be the tridiagonal matrix with $\alpha_{p}$ , $\beta_{p}$ , $\gamma_{p}$ as in (29.3.11), (29.3.12). Let the eigenvalues of $\mathbf{M}$ be $H_{p}$ with

29.15.3 $H_{0}<H_{1}<\cdots<H_{n},$

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Symbols:: : nonnegative integer and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and also let

29.15.4 $[A_{0},A_{2},\dots,A_{2n}]^{\mathrm{T}}$

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Symbols:: : nonnegative integer and $A_{2p}$ : coefficients
Referenced by:: §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

be the eigenvector corresponding to $H_{m}$ and normalized so that

29.15.5 $\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{n}A_{2p}^{2}=1$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $A_{2p}$ : coefficients
Referenced by:: §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and

29.15.6 $\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p}>0.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $A_{2p}$ : coefficients
Referenced by:: §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i), §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.7 $a^{2m}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

ⓘ

Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.1) applies, with $\phi$ again defined as in (29.2.5).

Polynomial $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$

ⓘ

See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n+1$ , $m=0,1,\dots,n$ , the Fourier series (29.6.16) terminates:

29.15.8 $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}A_{2p+1}\cos\left((2p% +1)\phi\right).$

ⓘ

Symbols:: $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $A_{2p}$ : coefficients
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

In (29.15.2) replace $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.3.13), (29.3.14). Also, replace (29.15.4), (29.15.5), (29.15.6) by

29.15.9 $[A_{1},A_{3},\dots,A_{2n+1}]^{\mathrm{T}},$

ⓘ

Symbols:: : nonnegative integer and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.15.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.10 $\sum_{p=0}^{n}A_{2p+1}^{2}=1,$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.15.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.11 $\sum_{p=0}^{n}A_{2p+1}>0.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.15.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.12 $a^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

ⓘ

Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.8) applies.

Polynomial $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$

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See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n+1$ , $m=0,1,\dots,n$ , the Fourier series (29.6.31) terminates:

29.15.13 $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}B_{2p+1}\sin\left((2p% +1)\phi\right).$

ⓘ

Symbols:: $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\sin\NVar{z}$ : sine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $B_{2p+1}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

In (29.15.2) replace $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.3.15), (29.3.16). Also, replace (29.15.4), (29.15.5), (29.15.6) by

29.15.14 $[B_{1},B_{3},\dots,B_{2n+1}]^{\mathrm{T}},$

ⓘ

Symbols:: : nonnegative integer and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.15 $\sum_{p=0}^{n}B_{2p+1}^{2}=1,$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.16 $\sum_{p=0}^{n}(2p+1)B_{2p+1}>0.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.17 $b^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

ⓘ

Symbols:: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.13) applies.

Polynomial $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$

ⓘ

See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n+1$ , $m=0,1,\dots,n$ , the Fourier series (29.6.8) terminates:

29.15.18 $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}\cos\left(2p\phi\right)\right).$

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $C_{2p+1}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

In (29.15.2) replace $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.6.11). Also, replace (29.15.4), (29.15.5), (29.15.6) by

29.15.19 $[C_{0},C_{2},\dots,C_{2n}]^{\mathrm{T}},$

ⓘ

Symbols:: : nonnegative integer and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.20 $\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{n}C_{2% p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{n-1}C_{2p}C_{2p+2}=1,$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer, : real parameter and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.21 $\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}>0.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.22 $a^{2m}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

ⓘ

Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.18) applies.

Polynomial $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$

ⓘ

See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n+2$ , $m=0,1,\dots,n$ , the Fourier series (29.6.46) terminates:

29.15.23 $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)=\sum_{p=0}^{n}B_{2p+2}\sin\left((2% p+2)\phi\right).$

ⓘ

Symbols:: $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\sin\NVar{z}$ : sine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $B_{2p+1}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

In (29.15.2) replace $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.3.17). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.24 $[B_{2},B_{4},\dots,B_{2n+2}]^{\mathrm{T}},$

ⓘ

Symbols:: : nonnegative integer and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.25 $\sum_{p=0}^{n}B_{2p+2}^{2}=1,$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.26 $\sum_{p=0}^{n}(2p+2)B_{2p+2}>0.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.27 $b^{2m+2}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

ⓘ

Symbols:: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.23) applies.

Polynomial $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$

ⓘ

See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n+2$ , $m=0,1,\dots,n$ , the Fourier series (29.6.23) terminates:

29.15.28 $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=0}^{n}C_{2p+1}\cos\left((2p+1)\phi\right).$

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\cos\NVar{z}$ : cosine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $C_{2p+1}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

In (29.15.2) replace $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.6.26). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.29 $[C_{1},C_{3},\dots,C_{2n+1}]^{\mathrm{T}},$

ⓘ

Symbols:: : nonnegative integer and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.30 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}C_{2p+1}^{2}-{\tfrac{1}{2}k^{2}% \left(\tfrac{1}{2}C_{1}^{2}+\sum_{p=0}^{n-1}C_{2p+1}C_{2p+3}\right)=1},$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer, : real parameter and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E30
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.31 $\sum_{p=0}^{n}C_{2p+1}>0.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.32 $a^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

ⓘ

Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.28) applies.

Polynomial $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$

ⓘ

See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n+2$ , $m=0,1,\dots,n$ , the Fourier series (29.6.38) terminates:

29.15.33 $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=0}^{n}D_{2p+1}\sin\left((2p+1)\phi\right).$

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\sin\NVar{z}$ : sine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $D_{2p}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

In (29.15.2) replace $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.6.41). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.34 $[D_{1},D_{3},\dots,D_{2n+1}]^{\mathrm{T}},$

ⓘ

Symbols:: : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.35 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+1}^{2}+{\tfrac{1}{2}k^{2}% \left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{n-1}D_{2p+1}D_{2p+3}\right)=1},$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer, : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.36 $\sum_{p=0}^{n}(2p+1)D_{2p+1}>0.$

ⓘ

Symbols:: : nonnegative integer, : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.37 $b^{2m+1}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

ⓘ

Symbols:: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.33) applies.

Polynomial $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$

ⓘ

See also:: Annotations for §29.15(i), §29.15 and Ch.29

When $\nu=2n+3$ , $m=0,1,\dots,n$ , the Fourier series (29.6.53) terminates:

29.15.38 $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=0}^{n}D_{2p+2}\sin\left((2p+2)\phi\right).$

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $\sin\NVar{z}$ : sine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $D_{2p}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

In (29.15.2) replace $\alpha_{p}$ , $\beta_{p}$ , and $\gamma_{p}$ as in (29.6.56). Also replace (29.15.4), (29.15.5), (29.15.6) by

29.15.39 $[D_{2},D_{4},\dots,D_{2n+2}]^{\mathrm{T}},$

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Symbols:: : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E39
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.40 $\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}% \sum_{p=1}^{n}D_{2p}D_{2p+2}=1,$

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Symbols:: : nonnegative integer, : nonnegative integer, : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E40
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

29.15.41 $\sum_{p=0}^{n}(2p+2)D_{2p+2}>0.$

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Symbols:: : nonnegative integer, : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E41
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

Then

29.15.42 $b^{2m+2}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$

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Symbols:: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E42
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

and (29.15.38) applies.

§29.15(ii) Chebyshev Series

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Keywords:: Chebyshev series, Lamé polynomials, explicit formulas
Notes:: See Arscott (1964b, §9.6.2) and Ince (1940a).
Referenced by:: §29.12(i)
Permalink:: http://dlmf.nist.gov/29.15.ii
See also:: Annotations for §29.15 and Ch.29

The Chebyshev polynomial of the first kind (§18.3) satisfies $\cos\left(p\phi\right)=T_{p}\left(\cos\phi\right)$ . Since (29.2.5) implies that $\cos\phi=\operatorname{sn}\left(z,k\right)$ , (29.15.1) can be rewritten in the form

29.15.43 $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p% }T_{2p}\left(\operatorname{sn}\left(z,k\right)\right).$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.15.E43
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

This determines the polynomial of degree for which $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=P({\operatorname{sn}}^{2}\left(z,k% \right))$ ; compare Table 29.12.1. The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an $(n+1)\times(n+1)$ tridiagonal matrix; see Arscott and Khabaza (1962).

Using also $\sin\left((p+1)\phi\right)=(\sin\phi)U_{p}\left(\cos\phi\right)$ , with denoting the Chebyshev polynomial of the second kind (§18.3), we obtain

29.15.44 $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}A_{2p+1}T_{2p+1}\left% (\operatorname{sn}\left(z,k\right)\right),$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $A_{2p}$ : coefficients
Permalink:: http://dlmf.nist.gov/29.15.E44
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

29.15.45 $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \sum_{p=0}^{n}B_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)\right),$

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E45
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

29.15.46 $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}T_{2p}\left(\operatorname{sn}\left% (z,k\right)\right)\right),$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $C_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E46
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

29.15.47 $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \sum_{p=0}^{n}B_{2p+2}U_{2p+1}\left(\operatorname{sn}\left(z,k\right)\right),$

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $B_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E47
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

29.15.48 $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=0}^{n}C_{2p+1}T_{2p+1}\left(\operatorname{sn}\left(z,k\right)\right),$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $C_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E48
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

29.15.49 $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+1}U_{2p}\left(% \operatorname{sn}\left(z,k\right)\right),$

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Symbols:: $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E49
Encodings:: TeX, pMML, png
See also:: Annotations for §29.15(ii), §29.15 and Ch.29

29.15.50 $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(% \operatorname{sn}\left(z,k\right)\right).$

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