►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities .
These solutions are the Heun polynomials.
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►Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)).
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►For exact values of the coefficients of the Jacobi polynomials
, the ultraspherical polynomials
, the Chebyshev polynomials
and , the Legendre polynomials
, the Laguerre polynomials
, and the Hermite polynomials
, see Abramowitz and Stegun (1964, pp. 793–801).
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►For another version of the discrete orthogonality property of the polynomials
see (3.11.9).
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►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)).
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►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)).
►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).
►For () see §14.33.
►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates , , , and for .
The ranges of are for and , and for and .
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►For , , and see §3.5(v).
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