§29.15 Fourier Series and Chebyshev Series

ⓘ

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

ⓘ

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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $n$ : 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		$H_{0}<H_{1}<\cdots<H_{n},$
ⓘ Symbols: $n$ : 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		$[A_{0},A_{2},\dots,A_{2n}]^{\mathrm{T}}$
ⓘ Symbols: $n$ : 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: $n$ : nonnegative integer, $p$ : 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: $n$ : nonnegative integer, $p$ : 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, $m$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $n$ : 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: $n$ : nonnegative integer, $p$ : 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: $n$ : nonnegative integer, $p$ : 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, $m$ : nonnegative integer, $k$ : 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 $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$

ⓘ

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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $n$ : 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: $n$ : nonnegative integer, $p$ : 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: $n$ : nonnegative integer, $p$ : 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, $m$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $n$ : 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: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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: $n$ : nonnegative integer, $p$ : 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, $m$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $n$ : 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: $n$ : nonnegative integer, $p$ : 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: $n$ : nonnegative integer, $p$ : 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, $m$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $n$ : 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: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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: $n$ : nonnegative integer, $p$ : 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, $m$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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: $n$ : 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: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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: $n$ : nonnegative integer, $p$ : 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, $m$ : nonnegative integer, $k$ : 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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		$[D_{2},D_{4},\dots,D_{2n+2}]^{\mathrm{T}},$
ⓘ Symbols: $n$ : 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		$\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}% \sum_{p=1}^{n}D_{2p}D_{2p+2}=1,$
ⓘ Symbols: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : 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		$\sum_{p=0}^{n}(2p+2)D_{2p+2}>0.$
ⓘ Symbols: $n$ : nonnegative integer, $p$ : 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		$b^{2m+2}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),$
ⓘ Symbols: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : 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

ⓘ

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 $T$ 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		$\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p% }T_{2p}\left(\operatorname{sn}\left(z,k\right)\right).$
ⓘ 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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 $P$ of degree $n$ 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 $U$ denoting the Chebyshev polynomial of the second kind (§18.3), we obtain

29.15.44	$\displaystyle\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$	$\displaystyle=\sum_{p=0}^{n}A_{2p+1}T_{2p+1}\left(\operatorname{sn}\left(z,k% \right)\right),$
ⓘ 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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	$\displaystyle\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{cn}\left(z,k\right)\sum_{p=0}^{n}B_{2p+1}U_{2p}% \left(\operatorname{sn}\left(z,k\right)\right),$
ⓘ 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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	$\displaystyle\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{dn}\left(z,k\right)\left(\tfrac{1}{2}C_{0}+\sum_{p% =1}^{n}C_{2p}T_{2p}\left(\operatorname{sn}\left(z,k\right)\right)\right),$
ⓘ 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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	$\displaystyle\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{cn}\left(z,k\right)\sum_{p=0}^{n}B_{2p+2}U_{2p+1}% \left(\operatorname{sn}\left(z,k\right)\right),$
ⓘ 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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	$\displaystyle\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}C_{2p+1}T_{2p+1}% \left(\operatorname{sn}\left(z,k\right)\right),$
ⓘ 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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	$\displaystyle\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k% \right)\sum_{p=0}^{n}D_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)% \right),$
ⓘ 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, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : 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	$\displaystyle\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$	$\displaystyle=\operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k% \right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(\operatorname{sn}\left(z,k\right)% \right).$
ⓘ 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, $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter and $D_{2p+1}$ : coefficents Permalink: http://dlmf.nist.gov/29.15.E50 Encodings: TeX, pMML, png See also: Annotations for §29.15(ii), §29.15 and Ch.29

For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).

§29.15 Fourier Series and Chebyshev Series

Contents

§29.15(i) Fourier Coefficients

Polynomial 𝑢𝐸2⁢nm⁡(z,k2)

Polynomial 𝑠𝐸2⁢n+1m⁡(z,k2)

Polynomial 𝑐𝐸2⁢n+1m⁡(z,k2)

Polynomial 𝑑𝐸2⁢n+1m⁡(z,k2)

Polynomial 𝑠𝑐𝐸2⁢n+2m⁡(z,k2)

Polynomial 𝑠𝑑𝐸2⁢n+2m⁡(z,k2)

Polynomial 𝑐𝑑𝐸2⁢n+2m⁡(z,k2)

Polynomial 𝑠𝑐𝑑𝐸2⁢n+3m⁡(z,k2)

§29.15(ii) Chebyshev Series

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

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

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

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

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

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

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

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