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1: 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 .
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 ,
2: Bibliography C
  • 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.
  • F. Chapeau-Blondeau and A. Monir (2002) Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.
  • Cunningham Project (website)
  • 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
    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
    For example, there are eight compositions of 4: 4 , 3 + 1 , 1 + 3 , 2 + 2 , 2 + 1 + 1 , 1 + 2 + 1 , 1 + 1 + 2 , and 1 + 1 + 1 + 1 . 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 , } . … 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
    26.5.1 C ( n ) = 1 n + 1 ( 2 n n ) = 1 2 n + 1 ( 2 n + 1 n ) = ( 2 n n ) - ( 2 n n - 1 ) = ( 2 n - 1 n ) - ( 2 n - 1 n + 1 ) .
    26.5.3 C ( n + 1 ) = k = 0 n C ( k ) C ( n - k ) ,
    26.5.4 C ( n + 1 ) = 2 ( 2 n + 1 ) n + 2 C ( n ) ,
    26.5.5 C ( n + 1 ) = k = 0 n / 2 ( n 2 k ) 2 n - 2 k C ( k ) .
    26.5.7 lim n C ( n + 1 ) C ( n ) = 4 .
    7: 27.18 Methods of Computation: Primes
    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
    24.15.6 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 ) ,
    24.15.8 k = 0 n ( - 1 ) n + k s ( n + 1 , k + 1 ) B k = n ! n + 1 .
    24.15.9 p B n n S ( p - 1 + n , p - 1 ) ( mod p 2 ) , 1 n p - 2 ,
    The Fibonacci numbers are defined by u 0 = 0 , u 1 = 1 , and u n + 1 = u n + u n - 1 , n 1 . …
    9: 3.6 Linear Difference Equations
    For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).
    10: 26.6 Other Lattice Path Numbers
    26.6.2 M ( n ) = k = 0 n ( - 1 ) k n + 2 - k ( n k ) ( 2 n + 2 - 2 k n + 1 - k ) .
    26.6.4 r ( n ) = D ( n , n ) - D ( n + 1 , n - 1 ) , n 1 .
    26.6.10 D ( m , n ) = D ( m , n - 1 ) + D ( m - 1 , n ) + D ( m - 1 , n - 1 ) , m , n 1 ,
    26.6.11 M ( n ) = M ( n - 1 ) + k = 2 n M ( k - 2 ) M ( n - k ) , n 2 .
    26.6.13 M ( n ) = k = 0 n ( - 1 ) k ( n k ) C ( n + 1 - k ) ,