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21: 26.2 Basic Definitions
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►Thus is the permutation , , .
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►As an example, is a partition of 13.
…See Table 26.2.1 for .
For the actual partitions () for see Table 26.4.1.
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►The example has six parts, three of which equal 1.
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22: 18.8 Differential Equations
23: 8.26 Tables
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Zhang and Jin (1996, Table 3.8) tabulates for , to 8D or 8S.
Zhang and Jin (1996, Table 3.9) tabulates for , , to 8D.
Chiccoli et al. (1988) presents a short table of for , to 14S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Stankiewicz (1968) tabulates for , to 7D.
24: 5.17 Barnes’ -Function (Double Gamma Function)
25: 26.9 Integer Partitions: Restricted Number and Part Size
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►The conjugate to the example in Figure 26.9.1 is .
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►It is also equal to the number of lattice paths from to that have exactly vertices , , , above and to the left of the lattice path.
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►It is also assumed everywhere that .
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►Also, when
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26: Bibliography G
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Fourier transforms related to a root system of rank 1.
Transform. Groups 12 (1), pp. 77–116.
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The solutions of Painlevé’s fifth equation.
Differ. Uravn. 12 (4), pp. 740–742 (Russian).
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One-parameter systems of solutions of Painlevé equations.
Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).
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Algorithm 300: Coulomb wave functions.
Comm. ACM 10 (4), pp. 244–245.
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27: 28.6 Expansions for Small
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►Leading terms of the of the power series for are:
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►The coefficients of the power series of , and also , are the same until the terms in and , respectively.
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►Numerical values of the radii of convergence of the power series (28.6.1)–(28.6.14) for are given in Table 28.6.1.
Here for , for , and for and .
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►where is the unique root of the equation in the interval , and .
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28: 28.26 Asymptotic Approximations for Large
29: 5.23 Approximations
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►Cody and Hillstrom (1967) gives minimax rational approximations for for the ranges , , ; precision is variable.
Hart et al. (1968) gives minimax polynomial and rational approximations to and in the intervals , , ; precision is variable.
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►Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of , , , , , and the first six derivatives of for .
…Clenshaw (1962) also gives 20D Chebyshev-series coefficients for and its reciprocal for .
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