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1: 28.6 Expansions for Small
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►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).
►The coefficients of the power series of , and also , are the same until the terms in and , respectively.
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►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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►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).
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2: 4.17 Special Values and Limits
3: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of and , , for (23D).
4: 24.2 Definitions and Generating Functions
5: 28.16 Asymptotic Expansions for Large
6: 28.15 Expansions for Small
7: 18.8 Differential Equations
8: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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is the number of permutations of with cycles of length 1, cycles of length 2, , and cycles of length :
… is the number of set partitions of with subsets of size 1, subsets of size 2, , and subsets of size :
…For each all possible values of are covered.
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►where the summation is over all nonnegative integers such that .
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9: 10.12 Generating Function and Associated Series
10: 24.19 Methods of Computation
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►If denotes the right-hand side of (24.19.1) but with the second product taken only for , then for .
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►For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11).
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►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
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