# Lamé polynomials

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##### 1: 29.14 Orthogonality
###### §29.14 Orthogonality
βΊLamé polynomials are orthogonal in two ways. … βΊ
29.14.4 $\mathit{sE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{sE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
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29.14.5 $\mathit{cE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{cE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
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29.14.6 $\mathit{dE}^{m}_{2n+1}\left(s,k^{2}\right)\mathit{dE}^{m}_{2n+1}\left(K+% \mathrm{i}t,k^{2}\right),$
##### 2: 29.12 Definitions
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###### §29.12(i) Elliptic-Function Form
βΊThere are eight types of Lamé polynomials, defined as follows: … βΊ
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##### 6: 29.20 Methods of Computation
βΊThese 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. … βΊ
###### §29.20(ii) LaméPolynomials
βΊThe corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. … βΊ
###### §29.20(iii) Zeros
βΊZeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …
##### 7: 29.16 Asymptotic Expansions
###### §29.16 Asymptotic Expansions
βΊHargrave 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
βΊAll derivatives are denoted by differentials, not by primes. βΊThe 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
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###### §29.22(ii) LaméPolynomials
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• LA3: Eigenvalues for Lamé polynomials.

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• LA4: Lamé polynomials.

• ##### 10: 29.21 Tables
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• 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.

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• 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.