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11: 27.17 Other Applications
§27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …
12: 24.10 Arithmetic Properties
§24.10 Arithmetic Properties
Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference.
§24.10(ii) Kummer Congruences
§24.10(iii) Voronoi’s Congruence
§24.10(iv) Factors
13: 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. Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts ± 2 ( mod 5 ) , partitions into parts ± 1 ( mod 5 ) , and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to [ 12 6 ] q . Goldberg et al. (1976) contains tables of binomial coefficients to n = 100 and Stirling numbers to n = 40 .
14: 24.14 Sums
§24.14 Sums
§24.14(i) Quadratic Recurrence Relations
24.14.6 k = 0 n ( n k ) 2 k B k E n - k = 2 ( 1 - 2 n - 1 ) B n - n E n - 1 .
§24.14(ii) Higher-Order Recurrence Relations
For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).
15: 26.1 Special Notation
( m n )

binomial coefficient.

m n

Eulerian number.

B ( n )

Bell number.

C ( n )

Catalan number.

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).
16: 27.1 Special Notation
§27.1 Special Notation
d , k , m , n

positive integers (unless otherwise indicated).

p , p 1 , p 2 ,

prime numbers (or primes): integers ( > 1 ) with only two positive integer divisors, 1 and the number itself.

x , y

real numbers.

17: 24.5 Recurrence Relations
§24.5 Recurrence Relations
24.5.3 k = 0 n - 1 ( n k ) B k = 0 , n = 2 , 3 , ,
24.5.5 k = 0 n ( n k ) 2 k E n - k + E n = 2 .
§24.5(ii) Other Identities
§24.5(iii) Inversion Formulas
18: 27.12 Asymptotic Formulas: Primes
§27.12 Asymptotic Formulas: Primes
Prime Number Theorem
The number of such primes not exceeding x is … There are infinitely many Carmichael numbers.
19: How to Cite
When referring to a specific item (chapter, section, equation, …), please mention the reference number. …Your readers can then readily locate the item within the book or online (at http://dlmf.nist.gov/5.4.E1 in this example; see Permalinks & Reference numbers). … Since the reference number is immediately recognized from the URL as Eq. …
Permalinks & Reference numbers
The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …
20: 24.6 Explicit Formulas
§24.6 Explicit Formulas
24.6.1 B 2 n = k = 2 2 n + 1 ( - 1 ) k - 1 k ( 2 n + 1 k ) j = 1 k - 1 j 2 n ,
24.6.2 B n = 1 n + 1 k = 1 n j = 1 k ( - 1 ) j j n ( n + 1 k - j ) / ( n k ) ,
24.6.9 B n = k = 0 n 1 k + 1 j = 0 k ( - 1 ) j ( k j ) j n ,
24.6.12 E 2 n = k = 0 2 n 1 2 k j = 0 k ( - 1 ) j ( k j ) ( 1 + 2 j ) 2 n .