Lamé polynomials

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1: 29.12 Definitions
§29.12(i) Elliptic-Function Form
There are eight types of Lamé polynomials, defined as follows: …
3: 29.14 Orthogonality
§29.14 Orthogonality
Lamé polynomials are orthogonal in two ways. …
29.14.4 $\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
29.14.5 $\mathit{cE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{cE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
29.14.6 $\mathit{dE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{dE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
4: 29.13 Graphics
§29.13(i) Eigenvalues for LaméPolynomials Figure 29.13.4: a 4 m ⁡ ( k 2 ) , b 4 m ⁡ ( k 2 ) as functions of k 2 for m = 0 , 1 , 2 , 3 , 4 ( a ’s), m = 1 , 2 , 3 , 4 ( b ’s). Magnify
§29.13(iii) LaméPolynomials: Complex Variable Figure 29.13.23: | uE 4 1 ⁡ ( x + i ⁢ y , 0.9 ) | for - 3 ⁢ K ⁡ ≤ x ≤ 3 ⁢ K ⁡ , 0 ≤ y ≤ 2 ⁢ K ′ ⁡ . … Magnify 3D Help
5: 29.20 Methods of Computation
These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. …
§29.20(ii) LaméPolynomials
The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
§29.20(iii) Zeros
Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …
6: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle $0\leq\Re z\leq K$, $0\leq\Im z\leq{K^{\prime}}$, when $nk$ and $nk^{\prime}$ assume large real values. …
8: 29.1 Special Notation
All derivatives are denoted by differentials, not by primes. The main functions treated in this chapter are the eigenvalues $a^{2m}_{\nu}\left(k^{2}\right)$, $a^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+1}_{\nu}\left(k^{2}\right)$, $b^{2m+2}_{\nu}\left(k^{2}\right)$, the Lamé functions $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$, $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)$, and the Lamé polynomials $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)$, $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)$, $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{sdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)$, $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)$. …
9: 29.22 Software
§29.22(ii) LaméPolynomials
• LA3: Eigenvalues for Lamé polynomials.

• LA4: Lamé polynomials.

• 10: 29.21 Tables
• Ince (1940a) tabulates the eigenvalues $a^{m}_{\nu}\left(k^{2}\right)$, $b^{m+1}_{\nu}\left(k^{2}\right)$ (with $a^{2m+1}_{\nu}$ and $b^{2m+1}_{\nu}$ interchanged) for $k^{2}=0.1,0.5,0.9$, $\nu=-\frac{1}{2},0(1)25$, and $m=0,1,2,3$. Precision is 4D.

• Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^{2}=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.