explicit formulas
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21: 18.3 Definitions
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3.
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►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii).
…Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for are given in §18.5(iv).
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►However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6).
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►Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23).
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As given by a Rodrigues formula (18.5.5).
22: 3.5 Quadrature
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Gauss–Legendre Formula
… ►Gauss–Chebyshev Formula
… ►Gauss–Laguerre Formula
… ►a complex Gauss quadrature formula is available. …23: 18.17 Integrals
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►Just as the indefinite integrals (18.17.1), (18.17.3) and (18.17.4), many similar formulas can be obtained by applying (1.4.26) to the differentiation formulas (18.9.15), (18.9.16) and (18.9.19)–(18.9.28).
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►Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of -th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively.
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►Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in (Erdélyi et al., 1953b, §10.9(15)).
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►Some of the resulting formulas are given below.
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►Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively.
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24: 14.15 Uniform Asymptotic Approximations
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►Provided that the corresponding expansions for and can be obtained from the connection formulas (14.9.7), (14.9.9), and (14.9.10).
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►For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000).
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►For asymptotic expansions and explicit error bounds, see Dunster (2003b).
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►For asymptotic expansions and explicit error bounds, see Olver (1997b, Chapter 12, §§12, 13) and Jones (2001).
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►For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986).
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25: 18.5 Explicit Representations
§18.5 Explicit Representations
… ►Chebyshev
… ►§18.5(ii) Rodrigues Formulas
… ►Related formula: … ►26: 2.5 Mellin Transform Methods
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►The inversion formula is given by
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►When , this identity is a Parseval-type formula; compare §1.14(iv).
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►This is allowable in view of the asymptotic formula
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►The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985).
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27: 13.2 Definitions and Basic Properties
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►Although does not exist when , , many formulas containing continue to apply in their limiting form.
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13.2.7
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13.2.8
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13.2.10
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§13.2(vii) Connection Formulas
…28: 18.39 Applications in the Physical Sciences
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►Here are three examples of solutions for (18.39.8) for explicit choices of and with the corresponding to the discrete spectrum.
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►Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below.
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►thus recapitulating, for , line 11 of Table 18.8.1, now shown with explicit normalization for the measure .
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►see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an in the denominator.
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