§29.15 Fourier Series and Chebyshev Series

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

29.15.1		$${\mathit{uE}}_{2n}^m(z,k^2)=\frac{1}{2}{A}_0+\sum _{p=1}^n{A}_{2p}\cos \left(2p\varphi \right).$$
ⓘ Symbols: ${\mathit{uE}}_{2n}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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 $p=1,2,\mathrm{\dots },n$, (29.6.3), and $A_{2n+2}=0$ can be cast as an algebraic eigenvalue problem in the following way. Let

29.15.2
ⓘ 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
ⓘ 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		$${[A_0,A_2,\mathrm{\dots },A_{2n}]}^{\mathrm{T}}$$
ⓘ 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		$$\frac{1}{2}{A}_0^2+\sum _{p=1}^n{A}_{2p}^2=1$$
ⓘ 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		$$\frac{1}{2}{A}_0+\sum _{p=1}^n{A}_{2p}>0.$$
ⓘ 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		$$a_{\nu }^{2m}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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 $\varphi$ again defined as in (29.2.5).

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

29.15.8		$${\mathit{sE}}_{2n+1}^m(z,k^2)=\sum _{p=0}^n{A}_{2p+1}\cos \left((2p+1)\varphi \right).$$
ⓘ Symbols: ${\mathit{sE}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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		$${[A_1,A_3,\mathrm{\dots },A_{2n+1}]}^{\mathrm{T}},$$
ⓘ 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		$$a_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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.

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

29.15.13		$${\mathit{cE}}_{2n+1}^m(z,k^2)=\sum _{p=0}^n{B}_{2p+1}\sin \left((2p+1)\varphi \right).$$
ⓘ Symbols: ${\mathit{cE}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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		$${[B_1,B_3,\mathrm{\dots },B_{2n+1}]}^{\mathrm{T}},$$
ⓘ 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		$$b_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $b_{\nu }^n\left(k^2\right)$: 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.

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

29.15.18		$${\mathit{dE}}_{2n+1}^m(z,k^2)=\mathrm{dn}(z,k)\left(\frac{1}{2}{C}_0+\sum _{p=1}^n{C}_{2p}\cos \left(2p\varphi \right)\right).$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathit{dE}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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		$${[C_0,C_2,\mathrm{\dots },C_{2n}]}^{\mathrm{T}},$$
ⓘ 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		$$\left(1-\frac{1}{2}{k}^2\right)\left(\frac{1}{2}{C}_0^2+\sum _{p=1}^n{C}_{2p}^2\right)-\frac{1}{2}{k}^2\sum _{p=0}^{n-1}{C}_{2p}{C}_{2p+2}=1,$$
ⓘ 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		$$\frac{1}{2}{C}_0+\sum _{p=1}^n{C}_{2p}>0.$$
ⓘ 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		$$a_{\nu }^{2m}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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.

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

29.15.23		$${\mathit{scE}}_{2n+2}^m(z,k^2)=\sum _{p=0}^n{B}_{2p+2}\sin \left((2p+2)\varphi \right).$$
ⓘ Symbols: ${\mathit{scE}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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		$${[B_2,B_4,\mathrm{\dots },B_{2n+2}]}^{\mathrm{T}},$$
ⓘ 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		$$b_{\nu }^{2m+2}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $b_{\nu }^n\left(k^2\right)$: 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.

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

29.15.28		$${\mathit{sdE}}_{2n+2}^m(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^n{C}_{2p+1}\cos \left((2p+1)\varphi \right).$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathit{sdE}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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		$${[C_1,C_3,\mathrm{\dots },C_{2n+1}]}^{\mathrm{T}},$$
ⓘ 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		$$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^n{C}_{2p+1}^2-\frac{1}{2}{k}^2\left(\frac{1}{2}{C}_1^2+\sum _{p=0}^{n-1}{C}_{2p+1}{C}_{2p+3}\right)=1,$$
ⓘ 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		$$a_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $a_{\nu }^n\left(k^2\right)$: 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.

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

29.15.33		$${\mathit{cdE}}_{2n+2}^m(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+1}\sin \left((2p+1)\varphi \right).$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathit{cdE}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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		$${[D_1,D_3,\mathrm{\dots },D_{2n+1}]}^{\mathrm{T}},$$
ⓘ 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		$$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^n{D}_{2p+1}^2+\frac{1}{2}{k}^2\left(\frac{1}{2}{D}_1^2-\sum _{p=0}^{n-1}{D}_{2p+1}{D}_{2p+3}\right)=1,$$
ⓘ 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		$$b_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $b_{\nu }^n\left(k^2\right)$: 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.

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

29.15.38		$${\mathit{scdE}}_{2n+3}^m(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+2}\sin \left((2p+2)\varphi \right).$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathit{scdE}}_{2n+3}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: 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		$${[D_2,D_4,\mathrm{\dots },D_{2n+2}]}^{\mathrm{T}},$$
ⓘ 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		$$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^n{D}_{2p+2}^2-\frac{1}{2}{k}^2\sum _{p=1}^n{D}_{2p}{D}_{2p+2}=1,$$
ⓘ 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		$$b_{\nu }^{2m+2}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$
ⓘ Symbols: $b_{\nu }^n\left(k^2\right)$: 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.