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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: 31.14 General Fuchsian Equation
The general second-order Fuchsian equation with N + 1 regular singularities at z = a j , j = 1 , 2 , , N , and at , is given by
31.14.1 d 2 w d z 2 + ( j = 1 N γ j z - a j ) d w d z + ( j = 1 N q j z - a j ) w = 0 , j = 1 N q j = 0 .
With a 1 = 0 and a 2 = 1 the total number of free parameters is 3 N - 3 . …
31.14.3 w ( z ) = ( j = 1 N ( z - a j ) - γ j / 2 ) W ( z ) ,
31.14.4 d 2 W d z 2 = j = 1 N ( γ ~ j ( z - a j ) 2 + q ~ j z - a j ) W , j = 1 N q ~ j = 0 ,
3: 31.15 Stieltjes Polynomials
31.15.2 j = 1 N γ j / 2 z k - a j + j = 1 j k n 1 z k - z j = 0 , k = 1 , 2 , , n .
31.15.6 a j < a j + 1 , j = 1 , 2 , , N - 1 ,
31.15.8 S m ( z 1 ) S m ( z 2 ) S m ( z N - 1 ) , z j ( a j , a j + 1 ) ,
4: 26.17 The Twelvefold Way
The twelvefold way gives the number of mappings f from set N of n objects to set K of k objects (putting balls from set N into boxes in set K ). …In this table ( k ) n is Pochhammer’s symbol, and S ( n , k ) and p k ( n ) are defined in §§26.8(i) and 26.9(i). …
Table 26.17.1: The twelvefold way.
elements of N elements of K f unrestricted f one-to-one f onto
labeled labeled k n ( k - n + 1 ) n k ! S ( n , k )
labeled unlabeled S ( n , 1 ) + S ( n , 2 ) + + S ( n , k ) { 1 n k 0 n > k S ( n , k )
5: 26.11 Integer Partitions: Compositions
c ( n ) denotes the number of compositions of n , and c m ( n ) is the number of compositions into exactly m parts. c ( T , n ) is the number of compositions of n with no 1’s, where again T = { 2 , 3 , 4 , } . …
26.11.1 c ( 0 ) = c ( T , 0 ) = 1 .
The Fibonacci numbers are determined recursively by … Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
6: 26.5 Lattice Paths: Catalan Numbers
§26.5 Lattice Paths: Catalan Numbers
§26.5(i) Definitions
C ( n ) is the Catalan number. …
§26.5(ii) Generating Function
§26.5(iii) Recurrence Relations
7: 27.18 Methods of Computation: Primes
§27.18 Methods of Computation: Primes
An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer x is given in Crandall and Pomerance (2005, §3.7). …An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … These algorithms are used for testing primality of Mersenne numbers, 2 n - 1 , and Fermat numbers, 2 2 n + 1 . …
8: 24.15 Related Sequences of Numbers
§24.15(i) Genocchi Numbers
§24.15(ii) Tangent Numbers
§24.15(iii) Stirling Numbers
§24.15(iv) Fibonacci and Lucas Numbers
The Fibonacci numbers are defined by u 0 = 0 , u 1 = 1 , and u n + 1 = u n + u n - 1 , n 1 . …
9: 26.6 Other Lattice Path Numbers
§26.6 Other Lattice Path Numbers
Delannoy Number D ( m , n )
Motzkin Number M ( n )
Narayana Number N ( n , k )
§26.6(iv) Identities
10: Bibliography C
  • L. Carlitz (1953) Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.
  • L. Carlitz (1954a) q -Bernoulli and Eulerian numbers. Trans. Amer. Math. Soc. 76 (2), pp. 332–350.
  • L. Carlitz (1954b) A note on Euler numbers and polynomials. Nagoya Math. J. 7, pp. 35–43.
  • J. R. Cash and R. V. M. Zahar (1994) A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.
  • Cunningham Project (website)