18.17.1 | |||
ⓘ
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18.17.2 | |||
ⓘ
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18.17.3 | |||
ⓘ
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18.17.4 | |||
ⓘ
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18.17.5 | |||
. | |||
ⓘ
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18.17.6 | |||
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For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977).
18.17.7 | |||
. | |||
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For the Ferrers function and Legendre function see §§14.3(i) and 14.3(ii), with and .
18.17.8 | |||
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For the parabolic cylinder function see §12.2. For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978).
18.17.9 | |||
, , | |||
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18.17.10 | ||||
, , | ||||
ⓘ
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18.17.11 | ||||
, , | ||||
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and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1.
18.17.12 | ||||
, , | ||||
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18.17.13 | ||||
, . | ||||
ⓘ
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18.17.14 | ||||
, . | ||||
ⓘ
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18.17.15 | ||||
. | ||||
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Throughout this subsection we assume ; sometimes however, this restriction can be eased by analytic continuation.
18.17.16 | |||
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For the beta function see §5.12, and for the confluent hypergeometric function see (16.2.1) and Chapter 13.
18.17.17 | |||
ⓘ
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18.17.18 | |||
ⓘ
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For the Bessel function see §10.2(ii).
18.17.19 | |||
ⓘ
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18.17.20 | |||
ⓘ
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18.17.21 | |||
ⓘ
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18.17.22 | |||
ⓘ
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18.17.23 | |||
ⓘ
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18.17.24 | |||
ⓘ
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18.17.25 | |||
ⓘ
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18.17.26 | |||
ⓘ
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18.17.27 | |||
ⓘ
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18.17.28 | |||
ⓘ
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18.17.29 | |||
ⓘ
|
18.17.30 | |||
ⓘ
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18.17.31 | |||
, , | |||
ⓘ
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18.17.32 | |||
, . | |||
ⓘ
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18.17.33 | |||
. | |||
ⓘ
|
For the confluent hypergeometric function see (16.2.1) and Chapter 13.
18.17.34 | |||
. | |||
ⓘ
|
18.17.35 | |||
. | |||
ⓘ
|
18.17.36 | |||
. | |||
ⓘ
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18.17.37 | |||
. | |||
ⓘ
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18.17.38 | |||
, | |||
ⓘ
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18.17.39 | |||
. | |||
ⓘ
|
18.17.40 | |||
, . | |||
ⓘ
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18.17.41 | |||
. Also, , even; , odd. | |||
ⓘ
|
For the generalized hypergeometric function see (16.2.1).
18.17.42 | |||
ⓘ
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18.17.43 | |||
ⓘ
|
These integrals are Cauchy principal values (§1.4(v)).
18.17.44 | |||
. | |||
ⓘ
|
The case is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972).
18.17.45 | |||
ⓘ
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18.17.46 | |||
ⓘ
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18.17.47 | |||
ⓘ
|
18.17.48 | |||
ⓘ
|
18.17.49 | |||
ⓘ
|
provided that is even and the sum of any two of is not less than the third; otherwise the integral is zero.
For further integrals, see Apelblat (1983, pp. 189–204), Erdélyi et al. (1954a, pp. 38–39, 94–95, 170–176, 259–261, 324), Erdélyi et al. (1954b, pp. 42–44, 271–294), Gradshteyn and Ryzhik (2000, pp. 788–806), Gröbner and Hofreiter (1950, pp. 23–30), Marichev (1983, pp. 216–247), Oberhettinger (1972, pp. 64–67), Oberhettinger (1974, pp. 83–92), Oberhettinger (1990, pp. 44–47 and 152–154), Oberhettinger and Badii (1973, pp. 103–112), Prudnikov et al. (1986b, pp. 420–617), Prudnikov et al. (1992a, pp. 419–476), and Prudnikov et al. (1992b, pp. 280–308).