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1: 20 Theta Functions
Chapter 20 Theta Functions
2: 26.13 Permutations: Cycle Notation
The Stirling cycle numbers of the first kind, denoted by [ n k ] , count the number of permutations of { 1 , 2 , , n } with exactly k cycles. They are related to Stirling numbers of the first kind by
26.13.3 [ n k ] = | s ( n , k ) | .
3: 26.8 Set Partitions: Stirling Numbers
§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 ( m n ) for m up to 50 and n up to 25; extends Table 26.4.1 to n = 10 ; tabulates Stirling numbers of the first and second kinds, s ( n , k ) and S ( n , k ) , for n up to 25 and k up to n ; tabulates partitions p ( n ) and partitions into distinct parts p ( 𝒟 , n ) for n up to 500. … It also contains a table of Gaussian polynomials up to [ 12 6 ] q . …
5: 26.14 Permutations: Order Notation
A descent of a permutation is a pair of adjacent elements for which the first is larger than the second. … In this subsection S ( n , k ) is again the Stirling number of the second kind26.8), and B m is the m th Bernoulli number24.2(i)). …
26.14.7 n k = j = 0 n k ( 1 ) n k j j ! ( n j k ) S ( n , j ) ,
26.14.12 S ( n , m ) = 1 m ! k = 0 n 1 n k ( k n m ) , n m , n 1 .
6: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
7: 26.1 Special Notation
x

real variable.

( m n )

binomial coefficient.

s ( n , k )

Stirling numbers of the first kind.

S ( n , k )

Stirling numbers of the second kind.

Other notations for s ( n , k ) , the Stirling numbers of the first kind, include S n ( k ) (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), S n k (Jordan (1939), Moser and Wyman (1958a)), ( n 1 k 1 ) B n k ( n ) (Milne-Thomson (1933)), ( 1 ) n k S 1 ( n 1 , n k ) (Carlitz (1960), Gould (1960)), ( 1 ) n k [ n k ] (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). Other notations for S ( n , k ) , the Stirling numbers of the second kind, include 𝒮 n ( k ) (Fort (1948)), 𝔖 n k (Jordan (1939)), σ n k (Moser and Wyman (1958b)), ( n k ) B n k ( k ) (Milne-Thomson (1933)), S 2 ( k , n k ) (Carlitz (1960), Gould (1960)), { n k } (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
8: 10.75 Tables
  • British Association for the Advancement of Science (1937) tabulates I 0 ( x ) , I 1 ( x ) , x = 0 ( .001 ) 5 , 7–8D; K 0 ( x ) , K 1 ( x ) , x = 0.01 ( .01 ) 5 , 7–10D; e x I 0 ( x ) , e x I 1 ( x ) , e x K 0 ( x ) , e x K 1 ( x ) , x = 5 ( .01 ) 10 ( .1 ) 20 , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of K 0 ( x ) , K 1 ( x ) for small values of x .

  • Bickley et al. (1952) tabulates x n I n ( x ) or e x I n ( x ) , x n K n ( x ) or e x K n ( x ) , n = 2 ( 1 ) 20 , x = 0 (.01 or .1) 10(.1) 20, 8S; I n ( x ) , K n ( x ) , n = 0 ( 1 ) 20 , x = 0 or 0.1 ( .1 ) 20 , 10S.

  • The main tables in Abramowitz and Stegun (1964, Chapter 9) give e x I n ( x ) , e x K n ( x ) , n = 0 , 1 , 2 , x = 0 ( .1 ) 10 ( .2 ) 20 , 8D–10D or 10S; x e x I n ( x ) , ( x / π ) e x K n ( x ) , n = 0 , 1 , 2 , 1 / x = 0 ( .002 ) 0.05 ; K 0 ( x ) + I 0 ( x ) ln x , x ( K 1 ( x ) I 1 ( x ) ln x ) , x = 0 ( .1 ) 2 , 8D; e x I n ( x ) , e x K n ( x ) , n = 3 ( 1 ) 9 , x = 0 ( .2 ) 10 ( .5 ) 20 , 5S; I n ( x ) , K n ( x ) , n = 0 ( 1 ) 20 ( 10 ) 50 , 100 , x = 1 , 2 , 5 , 10 , 50 , 100 , 9–10S.

  • Zhang and Jin (1996, pp. 240–250) tabulates I n ( x ) , I n ( x ) , K n ( x ) , K n ( x ) , n = 0 ( 1 ) 10 ( 10 ) 50 , 100 , x = 1 , 5 , 10 , 25 , 50 , 100 , 9S; I n + α ( x ) , I n + α ( x ) , K n + α ( x ) , K n + α ( x ) , n = 0 ( 1 ) 5 , 10, 30, 50, 100, α = 1 4 , 1 3 , 1 2 , 2 3 , 3 4 , x = 1 , 5, 10, 50, 8S; real and imaginary parts of I n + α ( z ) , I n + α ( z ) , K n + α ( z ) , K n + α ( z ) , n = 0 ( 1 ) 15 , 20(10)50, 100, α = 0 , 1 2 , z = 4 + 2 i , 20 + 10 i , 8S.

  • Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of K n ( z ) and K n ( z ) , for n = 2 ( 1 ) 20 , 9S.

  • 9: 24.15 Related Sequences of Numbers
    §24.15(iii) Stirling Numbers
    The Stirling numbers of the first kind s ( n , m ) , and the second kind S ( n , m ) , are as defined in §26.8(i).
    24.15.6 B n = k = 0 n ( 1 ) k k ! S ( n , k ) k + 1 ,
    24.15.7 B n = k = 0 n ( 1 ) k ( n + 1 k + 1 ) S ( n + k , k ) / ( n + k k ) ,
    24.15.9 p B n n S ( p 1 + n , p 1 ) ( mod p 2 ) , 1 n p 2 ,
    10: 27.2 Functions
    Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …
    Table 27.2.2: Functions related to division.
    n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
    5 4 2 6 18 6 6 39 31 30 2 32 44 20 6 84
    6 2 4 12 19 18 2 20 32 16 6 63 45 24 6 78
    7 6 2 8 20 8 6 42 33 20 4 48 46 22 4 72
    12 4 6 28 25 20 3 31 38 18 4 60 51 32 4 72