# §29.15 Fourier Series and Chebyshev Series

## §29.15(i) Fourier Coefficients

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

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

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

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

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

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

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

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

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

and (29.15.8) applies.

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

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

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

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

and (29.15.13) applies.

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

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

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

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

and (29.15.18) applies.

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

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

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

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

and (29.15.23) applies.

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

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

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

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

and (29.15.28) applies.

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

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

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

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

and (29.15.33) applies.

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

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

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

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

and (29.15.38) applies.

## §29.15(ii) Chebyshev Series

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

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 $\displaystyle\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$ $\displaystyle=\sum_{p=0}^{n}A_{2p+1}T_{2p+1}\left(\operatorname{sn}\left(z,k% \right)\right),$ 29.15.45 $\displaystyle\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$ $\displaystyle=\operatorname{cn}\left(z,k\right)\sum_{p=0}^{n}B_{2p+1}U_{2p}% \left(\operatorname{sn}\left(z,k\right)\right),$ 29.15.46 $\displaystyle\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$ $\displaystyle=\operatorname{dn}\left(z,k\right)\left(\tfrac{1}{2}C_{0}+\sum_{p% =1}^{n}C_{2p}T_{2p}\left(\operatorname{sn}\left(z,k\right)\right)\right),$ 29.15.47 $\displaystyle\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$ $\displaystyle=\operatorname{cn}\left(z,k\right)\sum_{p=0}^{n}B_{2p+2}U_{2p+1}% \left(\operatorname{sn}\left(z,k\right)\right),$ 29.15.48 $\displaystyle\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$ $\displaystyle=\operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}C_{2p+1}T_{2p+1}% \left(\operatorname{sn}\left(z,k\right)\right),$ 29.15.49 $\displaystyle\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$ $\displaystyle=\operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k% \right)\sum_{p=0}^{n}D_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)% \right),$ 29.15.50 $\displaystyle\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$ $\displaystyle=\operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k% \right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(\operatorname{sn}\left(z,k\right)% \right).$

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