With $\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k)$, as in (29.2.5), we have

29.6.1		$${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)=\frac{1}{2}{A}_0+\sum _{p=1}^{\mathrm{\infty }}{A}_{2p}\cos \left(2p\varphi \right).$$
ⓘ Symbols: ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients Referenced by: §29.15(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E1 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

Here

29.6.2		$$H=2a_{\nu }^{2m}\left(k^2\right)-\nu (\nu +1)k^2,$$
ⓘ Defines: $H$ (locally) Symbols: $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Referenced by: §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E2 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.3		$$({\beta }_0-H)A_0+{\alpha }_0{A}_2=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $H$ and $A_{2p}$: coefficients Referenced by: §29.15(i), §29.20(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E3 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.4		$${\gamma }_p{A}_{2p-2}+({\beta }_p-H)A_{2p}+{\alpha }_p{A}_{2p+2}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $H$ and $A_{2p}$: coefficients Referenced by: §29.15(i), §29.20(i), §29.6(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E4 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

with ${\alpha }_p$, ${\beta }_p$, and ${\gamma }_p$ as in (29.3.11) and (29.3.12), and

29.6.5		$$\frac{1}{2}{A}_0^2+\sum _{p=1}^{\mathrm{\infty }}{A}_{2p}^2=1,$$
ⓘ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Referenced by: §29.20(i), §29.6(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E5 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.6		$$\frac{1}{2}{A}_0+\sum _{p=1}^{\mathrm{\infty }}{A}_{2p}>0.$$
ⓘ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Referenced by: §29.20(i), §29.6(i), §29.6(i) Permalink: http://dlmf.nist.gov/29.6.E6 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

When $\nu \ne 2n$, where $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ the system (29.6.4)–(29.6.6) has a unique recessive solution $A_0,A_2,A_4,\mathrm{\dots }$; furthermore

29.6.7		$$\underset{p\to \mathrm{\infty }}{lim}\frac{A_{2p+2}}{A_{2p}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n$, or $\nu =2n$ and $m>n$.
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $A_{2p}$: coefficients Permalink: http://dlmf.nist.gov/29.6.E7 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies.

In the special case $\nu =2n$, $m=0,1,\mathrm{\dots },n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\mathrm{\dots }$. This solution can be constructed from (29.6.4) by backward recursion, starting with $A_{2n+2}=0$ and an arbitrary nonzero value of $A_{2n}$, followed by normalization via (29.6.5) and (29.6.6). Consequently, ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i).

An alternative version of the Fourier series expansion (29.6.1) is given by

29.6.8		$${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)=\mathrm{dn}(z,k)\left(\frac{1}{2}{C}_0+\sum _{p=1}^{\mathrm{\infty }}{C}_{2p}\cos \left(2p\varphi \right)\right).$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $C_{2p}$: coefficents Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E8 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

Here $\mathrm{dn}(z,k)$ is as in §22.2, and

29.6.9		$$({\beta }_0-H)C_0+{\alpha }_0{C}_2=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $H$ and $C_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E9 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.10		$${\gamma }_p{C}_{2p-2}+({\beta }_p-H)C_{2p}+{\alpha }_p{C}_{2p+2}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $H$ and $C_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E10 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

with ${\alpha }_p,{\beta }_p$, and ${\gamma }_p$ now defined by

29.6.11		${\alpha }_p$
		${\beta }_p$	$=4p^2(2-k^2),$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p+1)(\nu +2p)k^2,$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, ${\alpha }_p$, ${\beta }_p$ and ${\gamma }_p$ Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.6(i), §29.6 and Ch.29

and

29.6.12		$$\left(1-\frac{1}{2}{k}^2\right)\left(\frac{1}{2}{C}_0^2+\sum _{p=1}^{\mathrm{\infty }}{C}_{2p}^2\right)-\frac{1}{2}{k}^2\sum _{p=0}^{\mathrm{\infty }}{C}_{2p}{C}_{2p+2}=1,$$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $C_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E12 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.13		$$\frac{1}{2}{C}_0+\sum _{p=1}^{\mathrm{\infty }}{C}_{2p}>0,$$
ⓘ Symbols: $p$: nonnegative integer and $C_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E13 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.14		$$\underset{p\to \mathrm{\infty }}{lim}\frac{C_{2p+2}}{C_{2p}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n+1$, or $\nu =2n+1$ and $m>n$,
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $C_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E14 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.15		$$\frac{1}{2}{A}_0{C}_0+\sum _{p=1}^{\mathrm{\infty }}{A}_{2p}{C}_{2p}=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑐}}_{\nu }^{2m}(x,k^2)\right)}^{2}dx.$$
ⓘ Symbols: ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter, $\nu$: real parameter, $A_{2p}$: coefficients and $C_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E15 Encodings: TeX, pMML, png See also: Annotations for §29.6(i), §29.6 and Ch.29

29.6.16		$${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)=\sum _{p=0}^{\mathrm{\infty }}{A}_{2p+1}\cos \left((2p+1)\varphi \right).$$
ⓘ Symbols: ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E16 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

Here

29.6.17		$$H=2a_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2,$$
ⓘ Defines: $H$ (locally) Symbols: $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.6.E17 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.18		$$({\beta }_0-H)A_1+{\alpha }_0{A}_3=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $H$ and $A_{2p}$: coefficients Permalink: http://dlmf.nist.gov/29.6.E18 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.19		$${\gamma }_p{A}_{2p-1}+({\beta }_p-H)A_{2p+1}+{\alpha }_p{A}_{2p+3}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $H$ and $A_{2p}$: coefficients Permalink: http://dlmf.nist.gov/29.6.E19 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

with ${\alpha }_p$, ${\beta }_p$, and ${\gamma }_p$ as in (29.3.13) and (29.3.14), and

29.6.20		$$\sum _{p=0}^{\mathrm{\infty }}{A}_{2p+1}^2=1,$$
ⓘ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Referenced by: §29.6(ii) Permalink: http://dlmf.nist.gov/29.6.E20 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.21		$$\sum _{p=0}^{\mathrm{\infty }}{A}_{2p+1}>0,$$
ⓘ Symbols: $p$: nonnegative integer and $A_{2p}$: coefficients Permalink: http://dlmf.nist.gov/29.6.E21 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.22		$$\underset{p\to \mathrm{\infty }}{lim}\frac{A_{2p+1}}{A_{2p-1}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n+1$, or $\nu =2n+1$ and $m>n$.
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $A_{2p}$: coefficients Permalink: http://dlmf.nist.gov/29.6.E22 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

Also,

29.6.23		$${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^{\mathrm{\infty }}{C}_{2p+1}\cos \left((2p+1)\varphi \right),$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $C_{2p+1}$: coefficents Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E23 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

where

29.6.24		$$({\beta }_0-H)C_1+{\alpha }_0{C}_3=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $H$ and $C_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E24 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.25		$${\gamma }_p{C}_{2p-1}+({\beta }_p-H)C_{2p+1}+{\alpha }_p{C}_{2p+3}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $H$ and $C_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E25 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

with

29.6.26		${\alpha }_p$	$=\frac{1}{2}(\nu -2p-1)(\nu +2p+2)k^2,$
		${\beta }_p$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p)(\nu +2p+1)k^2,$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, ${\alpha }_p$, ${\beta }_p$ and ${\gamma }_p$ Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E26 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

and

29.6.27		$$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^{\mathrm{\infty }}{C}_{2p+1}^2-\frac{1}{2}{k}^2\left(\frac{1}{2}{C}_1^2+\sum _{p=0}^{\mathrm{\infty }}{C}_{2p+1}{C}_{2p+3}\right)=1,$$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $C_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E27 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.28		$$\sum _{p=0}^{\mathrm{\infty }}{C}_{2p+1}>0,$$
ⓘ Symbols: $p$: nonnegative integer and $C_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E28 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.29		$$\underset{p\to \mathrm{\infty }}{lim}\frac{C_{2p+1}}{C_{2p-1}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n+2$, or $\nu =2n+2$ and $m>n$,
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $C_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E29 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.30		$$\sum _{p=0}^{\mathrm{\infty }}{A}_{2p+1}{C}_{2p+1}=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(x,k^2)\right)}^{2}dx.$$
ⓘ Symbols: ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter, $\nu$: real parameter, $A_{2p}$: coefficients and $C_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E30 Encodings: TeX, pMML, png See also: Annotations for §29.6(ii), §29.6 and Ch.29

29.6.31		$${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)=\sum _{p=0}^{\mathrm{\infty }}{B}_{2p+1}\sin \left((2p+1)\varphi \right).$$
ⓘ Symbols: ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\sin z$: sine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E31 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

Here

29.6.32		$$H=2b_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2,$$
ⓘ Defines: $H$ (locally) Symbols: $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.6.E32 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.33		$$({\beta }_0-H)B_1+{\alpha }_0{B}_3=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $H$ and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E33 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.34		$${\gamma }_p{B}_{2p-1}+({\beta }_p-H)B_{2p+1}+{\alpha }_p{B}_{2p+3}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $H$ and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E34 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

with ${\alpha }_p$, ${\beta }_p$, and ${\gamma }_p$ as in (29.3.15), (29.3.16), and

29.6.35		$$\sum _{p=0}^{\mathrm{\infty }}{B}_{2p+1}^2=1,$$
ⓘ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Referenced by: §29.6(iii) Permalink: http://dlmf.nist.gov/29.6.E35 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.36		$$\sum _{p=0}^{\mathrm{\infty }}(2p+1)B_{2p+1}>0,$$
ⓘ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E36 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.37		$$\underset{p\to \mathrm{\infty }}{lim}\frac{B_{2p+1}}{B_{2p-1}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n+1$, or $\nu =2n+1$ and $m>n$.
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E37 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

Also,

29.6.38		$${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)=\mathrm{dn}(z,k)\sum _{p=0}^{\mathrm{\infty }}{D}_{2p+1}\sin \left((2p+1)\varphi \right),$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\sin z$: sine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $D_{2p+1}$: coefficents Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E38 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

where

29.6.39		$$({\beta }_0-H)D_1+{\alpha }_0{D}_3=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $D_{2p+1}$: coefficents and $H$ Permalink: http://dlmf.nist.gov/29.6.E39 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.40		$${\gamma }_p{D}_{2p-1}+({\beta }_p-H)D_{2p+1}+{\alpha }_p{D}_{2p+3}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $D_{2p+1}$: coefficents and $H$ Permalink: http://dlmf.nist.gov/29.6.E40 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

with

29.6.41		${\alpha }_p$	$=\frac{1}{2}(\nu -2p-1)(\nu +2p+2)k^2,$
		${\beta }_p$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p)(\nu +2p+1)k^2,$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, ${\alpha }_p$, ${\beta }_p$ and ${\gamma }_p$ Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E41 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

and

29.6.42		$$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=0}^{\mathrm{\infty }}{D}_{2p+1}^2+\frac{1}{2}{k}^2\left(\frac{1}{2}{D}_1^2-\sum _{p=0}^{\mathrm{\infty }}{D}_{2p+1}{D}_{2p+3}\right)=1,$$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $D_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E42 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.43		$$\sum _{p=0}^{\mathrm{\infty }}(2p+1)D_{2p+1}>0,$$
ⓘ Symbols: $p$: nonnegative integer and $D_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E43 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.44		$$\underset{p\to \mathrm{\infty }}{lim}\frac{D_{2p+1}}{D_{2p-1}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n+2$, or $\nu =2n+2$ and $m>n$,
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $D_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E44 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.45		$$\sum _{p=0}^{\mathrm{\infty }}{B}_{2p+1}{D}_{2p+1}=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(x,k^2)\right)}^{2}dx.$$
ⓘ Symbols: ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter, $\nu$: real parameter, $D_{2p+1}$: coefficents and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E45 Encodings: TeX, pMML, png See also: Annotations for §29.6(iii), §29.6 and Ch.29

29.6.46		$${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)=\sum _{p=1}^{\mathrm{\infty }}{B}_{2p}\sin \left(2p\varphi \right).$$
ⓘ Symbols: ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\sin z$: sine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E46 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

Here

29.6.47		$$H=2b_{\nu }^{2m+2}\left(k^2\right)-\nu (\nu +1)k^2,$$
ⓘ Defines: $H$ (locally) Symbols: $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter Permalink: http://dlmf.nist.gov/29.6.E47 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.48		$$({\beta }_0-H)B_2+{\alpha }_0{B}_4=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $H$ and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E48 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.49		$${\gamma }_p{B}_{2p}+({\beta }_p-H)B_{2p+2}+{\alpha }_p{B}_{2p+4}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $H$ and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E49 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

with ${\alpha }_p$, ${\beta }_p$, and ${\gamma }_p$ as in (29.3.17), and

29.6.50		$$\sum _{p=1}^{\mathrm{\infty }}{B}_{2p}^2=1,$$
ⓘ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Referenced by: §29.6(iv) Permalink: http://dlmf.nist.gov/29.6.E50 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.51		$$\sum _{p=0}^{\mathrm{\infty }}(2p+2)B_{2p+2}>0,$$
ⓘ Symbols: $p$: nonnegative integer and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E51 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.52		$$\underset{p\to \mathrm{\infty }}{lim}\frac{B_{2p+2}}{B_{2p}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n+2$, or $\nu =2n+2$ and $m>n$.
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $B_{2p+1}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E52 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

Also,

29.6.53		$${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)=\mathrm{dn}(z,k)\sum _{p=1}^{\mathrm{\infty }}{D}_{2p}\sin \left(2p\varphi \right),$$
ⓘ Symbols: $\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\sin z$: sine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $D_{2p}$: coefficents Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E53 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

where

29.6.54		$$({\beta }_0-H)D_2+{\alpha }_0{D}_4=0,$$
ⓘ Symbols: ${\alpha }_p$, ${\beta }_p$, $H$ and $D_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E54 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.55		$${\gamma }_p{D}_{2p}+({\beta }_p-H)D_{2p+2}+{\alpha }_p{D}_{2p+4}=0,$$
$p\ge 1$,
ⓘ Symbols: $p$: nonnegative integer, ${\alpha }_p$, ${\beta }_p$, ${\gamma }_p$, $H$ and $D_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E55 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

with

29.6.56		${\alpha }_p$	$=\frac{1}{2}(\nu -2p-2)(\nu +2p+3)k^2,$
		${\beta }_p$	$={(2p+2)}^2(2-k^2),$
		${\gamma }_p$	$=\frac{1}{2}(\nu -2p-1)(\nu +2p+2)k^2,$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, ${\alpha }_p$, ${\beta }_p$ and ${\gamma }_p$ Referenced by: §29.15(i) Permalink: http://dlmf.nist.gov/29.6.E56 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

and

29.6.57		$$\left(1-\frac{1}{2}{k}^2\right)\sum _{p=1}^{\mathrm{\infty }}{D}_{2p}^2-\frac{1}{2}{k}^2\sum _{p=1}^{\mathrm{\infty }}{D}_{2p}{D}_{2p+2}=1,$$
ⓘ Symbols: $p$: nonnegative integer, $k$: real parameter and $D_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E57 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.58		$$\sum _{p=0}^{\mathrm{\infty }}(2p+2)D_{2p+2}>0,$$
ⓘ Symbols: $p$: nonnegative integer and $D_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E58 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.59		$$\underset{p\to \mathrm{\infty }}{lim}\frac{D_{2p+2}}{D_{2p}}=\frac{k^2}{{(1+k^{\prime })}^2},$$
$\nu \ne 2n+3$, or $\nu =2n+3$ and $m>n$,
ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $D_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E59 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29

29.6.60		$$\sum _{p=1}^{\mathrm{\infty }}{B}_{2p}{D}_{2p}=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(x,k^2)\right)}^{2}dx.$$
ⓘ Symbols: ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$, $\int$: integral, $m$: nonnegative integer, $p$: nonnegative integer, $x$: real variable, $k$: real parameter, $\nu$: real parameter, $B_{2p+1}$: coefficents and $D_{2p}$: coefficents Permalink: http://dlmf.nist.gov/29.6.E60 Encodings: TeX, pMML, png See also: Annotations for §29.6(iv), §29.6 and Ch.29