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11: 33.20 Expansions for Small
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►where
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►As with and fixed,
…where is given by (33.14.11), (33.14.12), and
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►For a comprehensive collection of asymptotic expansions that cover and as and are uniform in , including unbounded values, see Curtis (1964a, §7).
These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders and .
12: 33.7 Integral Representations
13: 24.10 Arithmetic Properties
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►where , and is an arbitrary integer such that .
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►valid when and , where is a fixed integer.
…valid for fixed integers , and for all and such that .
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►valid for fixed integers , and for all such that
and .
…valid for fixed integers and for all such that .
14: 33.10 Limiting Forms for Large or Large
15: 10.42 Zeros
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►For example, if is real, then the zeros of are all complex unless for some positive integer , in which event has two real zeros.
►The distribution of the zeros of in the sector in the cases is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle so that in each case the cut lies along the positive imaginary axis.
The zeros in the sector are their conjugates.
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has no zeros in the sector ; this result remains true when is replaced by any real number
.
For the number of zeros of in the sector , when is real, see Watson (1944, pp. 511–513).
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16: 33.17 Recurrence Relations and Derivatives
17: 32.14 Combinatorics
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►Let be the group of permutations of the numbers
(§26.2).
With , is said to be an increasing
subsequence of of length
when .
Let be the length of the longest increasing subsequence of .
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32.14.1
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18: 28.23 Expansions in Series of Bessel Functions
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28.23.7
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28.23.9
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28.23.11
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28.23.13
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►When the series in the even-numbered equations converge for and , and the series in the odd-numbered equations converge for and .
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19: 26.1 Special Notation
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►Other notations for , the Stirling numbers of the first kind, include (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), (Jordan (1939), Moser and Wyman (1958a)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)).
►Other notations for , the Stirling numbers of the second kind, include (Fort (1948)), (Jordan (1939)), (Moser and Wyman (1958b)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
real variable. | |
nonnegative integers. | |
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binomial coefficient. | |
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Bell number. | |
Catalan number. | |
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