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βΊThese matrices are the same as those provided in §29.15(i) for the computation of Lamépolynomials with the difference that has to be chosen sufficiently large.
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βΊ
§29.20(ii) LaméPolynomials
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βΊThe corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamépolynomials.
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βΊ
§29.20(iii) Zeros
βΊZeros of Lamépolynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii).
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βΊHargrave and Sleeman (1977) give asymptotic approximations for Lamépolynomials and their eigenvalues, including error bounds.
The approximations for Lamépolynomials hold uniformly on the rectangle , , when and assume large real values.
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βΊAll derivatives are denoted by differentials, not by primes.
βΊThe main functions treated in this chapter are the eigenvalues , , , , the Lamé functions , , , , and the Lamépolynomials
, , , , , , , .
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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials
in
Table 29.12.1 (normalized so that the numerically largest
coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues for
, . Equations from §29.6 can be used
to transform to the normalization adopted in this chapter. Precision is 6S.