§29.3 Definitions and Basic Properties

Permalink:: http://dlmf.nist.gov/29.3

§29.3(i) Eigenvalues
§29.3(ii) Distribution
§29.3(iii) Continued Fractions
§29.3(iv) Lamé Functions
§29.3(v) Normalization
§29.3(vi) Orthogonality
§29.3(vii) Power Series

§29.3(i) Eigenvalues

Notes:: See Erdélyi et al. (1955, §15.5.1) and Magnus and Winkler (1966, §2.1).
Defines:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation and $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation
Keywords:: Lamé functions
Referenced by:: §28.34(ii), §29.3(iv)
Permalink:: http://dlmf.nist.gov/29.3.i

For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2\!\mathop{K\/}\nolimits\!$ or $4\!\mathop{K\/}\nolimits\!$ . They are denoted by $\mathop{a^{{2m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , where $m=0,1,2,\ldots$ ; see Table 29.3.1.

Table 29.3.1: Eigenvalues of Lamé’s equation.

eigenvalue $h$	parity	period
$\mathop{a^{{2m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	even	$2\!\mathop{K\/}\nolimits\!$
$\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	odd	$4\!\mathop{K\/}\nolimits\!$
$\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	even	$4\!\mathop{K\/}\nolimits\!$
$\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	odd	$2\!\mathop{K\/}\nolimits\!$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer, $h$ : real parameter, $k$ : real parameter and $\nu$ : real parameter
Keywords:: Lamé functions
Referenced by:: §29.3(i)
Permalink:: http://dlmf.nist.gov/29.3.T1

§29.3(ii) Distribution

Notes:: See Erdélyi (1941b) and Erdélyi et al. (1955, §15.5.1). For (29.3.6) see Volkmer (2004b).
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.ii

The eigenvalues interlace according to

29.3.1 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{a^{{m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right),$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E1
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29.3.2 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{b^{{m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right),$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E2
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29.3.3 $\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{b^{{m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right),$

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E3
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29.3.4 $\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{a^{{m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right).$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E4
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The eigenvalues coalesce according to

29.3.5 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)=\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right),$ $\nu=0,1,\dots,m-1$ .

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E5
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If $\nu$ is distinct from $0,1,\dots,m-1$ , then

29.3.6 $\left(\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)\right)\nu(\nu-1)\cdots(\nu-m+1)>0.$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Referenced by:: §29.3(ii)
Permalink:: http://dlmf.nist.gov/29.3.E6
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If $\nu$ is a nonnegative integer, then

29.3.7 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)+\mathop{a^{{\nu-m}}_{{\nu}}\/}\nolimits\!\left(1-k^{2}\right)=\nu(\nu+1),$ $m=0,1,\dots,\nu$ ,

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E7
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29.3.8 ${\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)+\mathop{b^{{\nu-m+1}}_{{\nu}}\/}\nolimits\!\left(1-k^{2}\right)=\nu(\nu+1)},$ $m=1,2,\dots,\nu$ .

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E8
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For the special case $k=k^{{\prime}}=\ifrac{1}{\sqrt{2}}$ see Erdélyi et al. (1955, §15.5.2).

§29.3(iii) Continued Fractions

Notes:: See Ince (1940b).
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.iii

The quantity

29.3.9 $H=2\!\mathop{a^{{2m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $H$
Referenced by:: §29.3(iii)
Permalink:: http://dlmf.nist.gov/29.3.E9
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satisfies the continued-fraction equation

29.3.10 $\beta _{p}-H-\cfrac{\alpha _{{p-1}}\gamma _{p}}{\beta _{{p-1}}-H-\cfrac{\alpha _{{p-2}}\gamma _{{p-1}}}{\beta _{{p-2}}-H-}}\dots=\cfrac{\alpha _{p}\gamma _{{p+1}}}{\beta _{{p+1}}-H-\cfrac{\alpha _{{p+1}}\gamma _{{p+2}}}{\beta _{{p+2}}-H-}}\cdots,$

Symbols:: $p$ : nonnegative integer, $H$ , $\alpha _{p}$ , $\beta _{p}$ and $\gamma _{p}$
Referenced by:: §29.20(i), §29.3(iii), §29.3(iii), §29.3(iii), §29.3(iii)
Permalink:: http://dlmf.nist.gov/29.3.E10
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where $p$ is any nonnegative integer, and

29.3.11 $\alpha _{p}=\begin{cases}(\nu-1)(\nu+2)k^{2},&p=0,\\ \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},&p\geq 1,\end{cases}$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $\alpha _{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.3.E11
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29.3.12

$\beta _{p}=4p^{2}(2-k^{2}),$

$\gamma _{p}=\tfrac{1}{2}(\nu-2p+2)(\nu+2p-1)k^{2}.$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter, $\beta _{p}$ and $\gamma _{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.3.E12
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The continued fraction following the second negative sign on the left-hand side of (29.3.10) is finite: it equals 0 if $p=0$ , and if $p>0$ , then the last denominator is $\beta _{0}-H$ . If $\nu$ is a nonnegative integer and $2p\leq\nu$ , then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with $2m\leq\nu$ . If $\nu$ is a nonnegative integer and $2p>\nu$ , then (29.3.10) has only the solutions (29.3.9) with $2m>\nu$ .

The quantity $H=2\!\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

29.3.13 $\beta _{p}=\begin{cases}2-k^{2}+\tfrac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $\beta _{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(ii)
Permalink:: http://dlmf.nist.gov/29.3.E13
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29.3.14

$\alpha _{p}=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$

$\gamma _{p}=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}.$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter, $\alpha _{p}$ and $\gamma _{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(ii)
Permalink:: http://dlmf.nist.gov/29.3.E14
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The quantity $H=2\!\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

29.3.15 $\beta _{p}=\begin{cases}2-k^{2}-\tfrac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $\beta _{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(iii)
Permalink:: http://dlmf.nist.gov/29.3.E15
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29.3.16

$\alpha _{p}=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$

$\gamma _{p}=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}.$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter, $\alpha _{p}$ and $\gamma _{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(iii)
Permalink:: http://dlmf.nist.gov/29.3.E16
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The quantity $H=2\!\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

29.3.17

$\alpha _{p}=\tfrac{1}{2}(\nu-2p-3)(\nu+2p+4)k^{2},$

$\beta _{p}=(2p+2)^{2}(2-k^{2}),$

$\gamma _{p}=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}.$

Symbols:: $p$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter, $\alpha _{p}$ , $\beta _{p}$ and $\gamma _{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(iv)
Permalink:: http://dlmf.nist.gov/29.3.E17
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§29.3(iv) Lamé Functions

Notes:: See Erdélyi et al. (1955, §15.5.1).
Defines:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function and $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.iv

The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by $\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ , $\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ , $\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ , $\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ . They are called Lamé functions with real periods and of order $\nu$ , or more simply, Lamé functions. See Table 29.3.2. In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,\!\mathop{K\/}\nolimits\!)$ , whereas the superscripts $2m$ , $2m+1$ , or $2m+2$ correspond to the number of zeros in $[0,2\!\mathop{K\/}\nolimits\!)$ .

Table 29.3.2: Lamé functions.

boundary conditions

eigenvalue

$h$

eigenfunction

$w(z)$

parity of

$w(z)$

parity of

$w(z-\!\mathop{K\/}\nolimits\!)$

period of

$w(z)$

$\left.\ifrac{dw}{dz}\right|_{{z=0}}=\left.\ifrac{dw}{dz}\right|_{{z=\!\mathop{K\/}\nolimits\!}}=0$

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

$\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

even

$2\!\mathop{K\/}\nolimits\!$

$w(0)=\left.\ifrac{dw}{dz}\right|_{{z=\!\mathop{K\/}\nolimits\!}}=0$

$\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$

$\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

odd

even

$4\!\mathop{K\/}\nolimits\!$

$\left.\ifrac{dw}{dz}\right|_{{z=0}}=w(\!\mathop{K\/}\nolimits\!)=0$

$\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$

$\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

even

odd

$4\!\mathop{K\/}\nolimits\!$

$w(0)=w(\!\mathop{K\/}\nolimits\!)=0$

$\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$

$\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

odd

$2\!\mathop{K\/}\nolimits\!$

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $m$ : nonnegative integer, $z$ : complex variable, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : eigenfunction
Keywords:: Lamé functions, Lamé’s equation
Referenced by:: §29.3(iv)
Permalink:: http://dlmf.nist.gov/29.3.T2

§29.3(v) Normalization

Notes:: See Jansen (1977, §3.1).
Keywords:: Lamé functions
Referenced by:: §29.20(i)
Permalink:: http://dlmf.nist.gov/29.3.v

29.3.18

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)\left(\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\right)^{2}dx=\frac{1}{4}\pi,$

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)\left(\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\right)^{2}dx=\frac{1}{4}\pi,$

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)\left(\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\right)^{2}dx=\frac{1}{4}\pi,$

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k\right)\left(\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\right)^{2}dx=\frac{1}{4}\pi.$

For $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ see §22.2.

To complete the definitions, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!,k^{2}\right)$ is positive and $\left.\ifrac{d\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)}{dz}\right|_{{z=\!\mathop{K\/}\nolimits\!}}$ is negative.

§29.3(vi) Orthogonality

Notes:: Combine Hochstadt (1964, p. 148) and Table 29.3.2.
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.vi

For $m\neq p$ ,

29.3.19

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Ec}^{{2p}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)dx=0,$

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Ec}^{{2p+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)dx=0,$

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Es}^{{2p+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)dx=0,$

$\int _{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Es}^{{2p+2}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}\right)dx=0.$