# §29.12 Definitions

## §29.12(i) Elliptic-Function Form

Throughout §§29.1229.16 the order $\nu$ in the differential equation (29.2.1) is assumed to be a nonnegative integer.

The Lamé functions $\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)$, $m=0,1,\dots,\nu$, and $\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)$, $m=1,2,\dots,\nu$, are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows:

 29.12.1 $\displaystyle\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Ec}^{2m}_{2n}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{uE}^{\NVar{m}}_{2\NVar{n}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Referenced by: §29.12(i) Permalink: http://dlmf.nist.gov/29.12.E1 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29 29.12.2 $\displaystyle\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Ec}^{2m+1}_{2n+1}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Permalink: http://dlmf.nist.gov/29.12.E2 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29 29.12.3 $\displaystyle\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Es}^{2m+1}_{2n+1}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{cE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Permalink: http://dlmf.nist.gov/29.12.E3 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29 29.12.4 $\displaystyle\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Ec}^{2m}_{2n+1}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{dE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Permalink: http://dlmf.nist.gov/29.12.E4 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29 29.12.5 $\displaystyle\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Es}^{2m+2}_{2n+2}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Permalink: http://dlmf.nist.gov/29.12.E5 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29 29.12.6 $\displaystyle\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Ec}^{2m+1}_{2n+2}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Permalink: http://dlmf.nist.gov/29.12.E6 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29 29.12.7 $\displaystyle\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Es}^{2m+1}_{2n+2}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Permalink: http://dlmf.nist.gov/29.12.E7 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29 29.12.8 $\displaystyle\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$ $\displaystyle=\mathit{Es}^{2m+2}_{2n+3}\left(z,k^{2}\right),$ ⓘ Defines: $\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial Symbols: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter Referenced by: §29.12(i) Permalink: http://dlmf.nist.gov/29.12.E8 Encodings: TeX, pMML, png See also: Annotations for §29.12(i), §29.12 and Ch.29

where $n=0,1,2,\dots$, $m=0,1,2,\dots,n$. These functions are polynomials in $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, and $\operatorname{dn}\left(z,k\right)$. In consequence they are doubly-periodic meromorphic functions of $z$.

The superscript $m$ on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of $z$-zeros of each Lamé polynomial in the interval $(0,K)$, while $n-m$ is the number of $z$-zeros in the open line segment from $K$ to $K+\mathrm{i}{K^{\prime}}$.

The prefixes $\mathit{u}$, $\mathit{s}$, $\mathit{c}$, $\mathit{d}$, $\mathit{sc}$, $\mathit{sd}$, $\mathit{cd}$, $\mathit{scd}$ indicate the type of the polynomial form of the Lamé polynomial; compare the 3rd and 4th columns in Table 29.12.1. In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\operatorname{sn}}^{2})$ denotes a polynomial of degree $n$ in ${\operatorname{sn}}^{2}\left(z,k\right)$ (different for each type). For the determination of the coefficients of the $P$’s see §29.15(ii).

## §29.12(ii) Algebraic Form

With the substitution $\xi={\operatorname{sn}}^{2}\left(z,k\right)$ every Lamé polynomial in Table 29.12.1 can be written in the form

 29.12.9 $\xi^{\rho}(\xi-1)^{\sigma}(\xi-k^{-2})^{\tau}P(\xi),$

where $\rho$, $\sigma$, $\tau$ are either $0$ or $\frac{1}{2}$. The polynomial $P(\xi)$ is of degree $n$ and has $m$ zeros (all simple) in $(0,1)$ and $n-m$ zeros (all simple) in $(1,k^{-2})$. The functions (29.12.9) satisfy (29.2.2).

## §29.12(iii) Zeros

Let $\xi_{1},\xi_{2},\dots,\xi_{n}$ denote the zeros of the polynomial $P$ in (29.12.9) arranged according to

 29.12.10 $0<\xi_{1}<\cdots<\xi_{m}<1<\xi_{m+1}<\cdots<\xi_{n} ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer, $k$: real parameter and $\xi_{n}$: zeros Permalink: http://dlmf.nist.gov/29.12.E10 Encodings: TeX, pMML, png See also: Annotations for §29.12(iii), §29.12 and Ch.29

Then the function

 29.12.11 $g(t_{1},t_{2},\dots,t_{n})=\left(\prod_{p=1}^{n}t_{p}^{\rho+\frac{1}{4}}|t_{p}% -1|^{\sigma+\frac{1}{4}}(k^{-2}-t_{p})^{\tau+\frac{1}{4}}\right)\prod_{q ⓘ Defines: $g(t_{1},\ldots,t_{n})$: function (locally) Symbols: $n$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, ($0$ or $\tfrac{1}{2}$): parameter $\rho$, ($0$ or $\tfrac{1}{2}$): parameter $\sigma$ and ($0$ or $\tfrac{1}{2}$): parameter $\tau$ Referenced by: §29.20(iii) Permalink: http://dlmf.nist.gov/29.12.E11 Encodings: TeX, pMML, png See also: Annotations for §29.12(iii), §29.12 and Ch.29

defined for $(t_{1},t_{2},\dots,t_{n})$ with

 29.12.12 $0\leq t_{1}\leq\cdots\leq t_{m}\leq 1\leq t_{m+1}\leq\cdots\leq t_{n}\leq k^{-% 2},$ ⓘ Symbols: $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter Referenced by: §29.12(iii) Permalink: http://dlmf.nist.gov/29.12.E12 Encodings: TeX, pMML, png See also: Annotations for §29.12(iii), §29.12 and Ch.29

attains its absolute maximum iff $t_{j}=\xi_{j}$, $j=1,2,\dots,n$. Moreover,

 29.12.13 ${\frac{\rho+\frac{1}{4}}{\xi_{p}}+\frac{\sigma+\frac{1}{4}}{\xi_{p}-1}+\frac{% \tau+\frac{1}{4}}{\xi_{p}-k^{-2}}+\sum_{\begin{subarray}{c}q=1\\ q\neq p\end{subarray}}^{n}\frac{1}{\xi_{p}-\xi_{q}}=0},$ $p=1,2,\dots,n$.

This result admits the following electrostatic interpretation: Given three point masses fixed at $t=0$, $t=1$, and $t=k^{-2}$ with positive charges $\rho+\tfrac{1}{4}$, $\sigma+\tfrac{1}{4}$, and $\tau+\tfrac{1}{4}$, respectively, and $n$ movable point masses at $t_{1},t_{2},\dots,t_{n}$ arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when $t_{j}=\xi_{j}$ for $j=1,2,\dots,n$.