29 Lamé Functions Lamé Polynomials 29.14 Orthogonality 29.16 Asymptotic Expansions

§29.15 Fourier Series and Chebyshev Series

Keywords:: Fourier series, Lamé polynomials
Permalink:: http://dlmf.nist.gov/29.15

§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

¶ Polynomial $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$

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

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

Symbols:: $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{\cos\/}\nolimits z$ : 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

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

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 $H_{0}<H_{1}<\dots<H_{n},$

Symbols:: : nonnegative integer and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E3
Encodings:: TeX, pMML, png

and also let

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

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

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

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

Then

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

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

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

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

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

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

Symbols:: $\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{\cos\/}\nolimits z$ : 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

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

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

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

Then

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

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

and (29.15.8) applies.

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

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

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

Symbols:: $\mathop{\mathit{cE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{\sin\/}\nolimits z$ : 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

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

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

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

Then

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

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

and (29.15.13) applies.

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

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

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

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

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 $C_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E19
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer, : nonnegative integer, : real parameter and $C_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E20
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer, : nonnegative integer and $C_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E21
Encodings:: TeX, pMML, png

Then

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

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

and (29.15.18) applies.

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

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

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

Symbols:: $\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{\sin\/}\nolimits z$ : 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

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

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

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

Then

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

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

and (29.15.23) applies.

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

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

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

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{\cos\/}\nolimits z$ : 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

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

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

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

Then

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

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

and (29.15.28) applies.

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

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

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

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{cdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, $\phi$ : change of variable and $D_{{2p+1}}$ : coefficents
Referenced by:: ¶ ‣ §29.15(i)
Permalink:: http://dlmf.nist.gov/29.15.E33
Encodings:: TeX, pMML, png

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 $D_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E34
Encodings:: TeX, pMML, png

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 $D_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E35
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer, : nonnegative integer and $D_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E36
Encodings:: TeX, pMML, png

Then

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

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

and (29.15.33) applies.

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

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

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

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{\sin\/}\nolimits z$ : 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

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:: : nonnegative integer and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E39
Encodings:: TeX, pMML, png

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:: : nonnegative integer, : nonnegative integer, : real parameter and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E40
Encodings:: TeX, pMML, png

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

Symbols:: : nonnegative integer, : nonnegative integer and $D_{{2p}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E41
Encodings:: TeX, pMML, png

Then

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

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : 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

and (29.15.38) applies.

§29.15(ii) Chebyshev Series

Notes:: See Arscott (1964b, §9.6.2) and Ince (1940a).
Keywords:: Lamé polynomials
Referenced by:: §29.12(i)
Permalink:: http://dlmf.nist.gov/29.15.ii

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

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : 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

This determines the polynomial of degree for which $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)=P({\mathop{% \mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right))$ ; 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 $\mathop{\sin\/}\nolimits\!\left((p+1)\phi\right)=(\mathop{\sin\/}\nolimits\phi% )\mathop{U_{{p}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\phi\right)$ , with $\mathop{U\/}\nolimits$ denoting the Chebyshev polynomial of the second kind (§18.3), we obtain

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : 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

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

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{cE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : 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

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{dE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : 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

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

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : 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

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : 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

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

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{cdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : 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

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

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter and $D_{{2p+1}}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E50
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 29.14 Orthogonality 29.16 Asymptotic Expansions

§29.15 Fourier Series and Chebyshev Series

Contents

§29.15(i) Fourier Coefficients

¶ Polynomial

¶ Polynomial

¶ Polynomial

¶ Polynomial

¶ Polynomial

¶ Polynomial

¶ Polynomial

¶ Polynomial

§29.15(ii) Chebyshev Series

¶ Polynomial $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$

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

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

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

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

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

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

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