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1: 20 Theta Functions
Chapter 20 Theta Functions
…2: 26.13 Permutations: Cycle Notation
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►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
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26.13.3
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3: 26.8 Set Partitions: Stirling Numbers
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§26.8(i) Definitions
… ► … ►§26.8(ii) Generating Functions
… ►§26.8(iv) Recurrence Relations
… ►§26.8(v) Identities
…4: 26.21 Tables
§26.21 Tables
►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions and partitions into distinct parts for up to 500. … ►It also contains a table of Gaussian polynomials up to . …5: 26.14 Permutations: Order Notation
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►A descent of a permutation is a pair of adjacent elements for which the first is larger than the second.
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►In this subsection is again the Stirling number of the second kind (§26.8), and is the th Bernoulli number (§24.2(i)).
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26.14.7
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26.14.12
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6: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…7: 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. | |
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binomial coefficient. | |
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Stirling numbers of the first kind. | |
Stirling numbers of the second kind. |
8: 10.75 Tables
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British Association for the Advancement of Science (1937) tabulates , , , 7–8D; , , , 7–10D; , , , , , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of , for small values of .
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
The main tables in Abramowitz and Stegun (1964, Chapter 9) give , , , , 8D–10D or 10S; , , , ; , , , 8D; , , , , 5S; , , , , 9–10S.
Zhang and Jin (1996, pp. 240–250) tabulates , , , , , , 9S; , , , , , 10, 30, 50, 100, , , , , , , 5, 10, 50, 8S; real and imaginary parts of , , , , , 20(10)50, 100, , , 8S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
9: Bibliography G
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Algorithm 236: Bessel functions of the first kind.
Comm. ACM 7 (8), pp. 479–480.
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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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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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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