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error generating summary
3: Bibliography C
  • CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
  • 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)
  • 4: 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 ,
    5: Browsers
    If none of those solutions work for you, you may explicitly choose a format such as HTML+images, using images for mathematics, at Customize DLMF. …
    6: 3.6 Linear Difference Equations
    For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).
    7: 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 ) ,
    8: Viewing DLMF Interactive 3D Graphics
    After installing the viewer you must select Customize DLMF on the DLMF Menu bar and choose either VRML or X3D under “Visualization Format. …
    9: 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 )
    10: 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).