Bernoulli_A — Number Theory

Calculates array of Bernoulli numbers using recurrent relations.

Interface

#include <codecogs/maths/discrete/number_theory/bernoulli_a.h>

using namespace Maths::Discrete::Number_Theory;

void bernoulli_A( int iMax, double* dB )

Given starting values with indices 0 and 1, this function calculates all numbers with indices up to some maximal index. The relations used are as follows:

$$B_0=1$$

(1)

$$B_1=-\frac{1}{2}$$

(2)

$$\frac{2n+1}{2n} B_{2n} = B_1 B_{2n-1} - \sum_{r=0}^{n-2} \frac{1}{2r+2} \big ( \begin{array}{l} 2n-1 \\ 2r+1 \end{array} \big ) B_{2r+2} B_{2n-2r-2}$$

(3)

$$B_{2n+1}=0, n=1, 2,...$$

(4)

$$B_{2n}= -\frac{2n}{2n+1} \sum_{r=0}^{n-2} \frac{1}{2r+2} \left ( \begin{array}{c} 2n-1 \\ 2r+1 \end{array} \right) B_{2r+2} B_{2n-2r-2}, n=2, 3,...$$

(5)

First numbers are:

$$\begin{array}{l} B_0 \ = \ 1 \\ B_1 \ = \ -\frac{1}{2} \\ B_2 \ = \ \frac{1}{6} \\ B_4 \ = \ -\frac{1}{30} \\ B_6 \ = \ \frac{1}{42} \\ B_8 \ = \ -\frac{1}{30} \\ B_{10} \ = \ \frac{5}{66} \\ B_{12} \ = \ -\frac{691}{2,730} \\ B_{14} \ = \ \frac{7}{6} \\ B_{16} \ = \ -\frac{3,617}{510} \\ B_{18} \ = \ \frac{43,867}{798} \\ B_{20} \ = \ -\frac{174,611}{330} \\ B_{22} \ = \ \frac{854,513}{138} \end{array}$$

(6)

References

Higher Transcendental Functions, vol.1, (1.13) by H.Bateman and A.Erdelyi (Bateman Manuscript Project), 1953
M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, 1964 chapt.23
Yu.Luke, Mathematical functions and their approximations, 1975 chapt.14.2

Example 1

#include <stdio.h>
#include <codecogs/maths/discrete/number_theory/bernoulli_a.h>

#define MAX_INDEX 16

int main()
{
  double dBernoulli[MAX_INDEX+1];

  printf( "%8s%2c%20s\n", " ", 'n', "Bn" );
  printf( "%8s", " " );
  for(int i = 0; i < 22; i++ )
    printf( "%c", '-' );
  printf( "\n" );

  Maths::Discrete::Number_Theory::bernoulli_A( MAX_INDEX, dBernoulli );

  printf( "%10d%20.12f\n", 0, dBernoulli[0] );
  printf( "%10d%20.12f\n", 1, dBernoulli[1] );
  for(int i = 2; i <= MAX_INDEX; i += 2 )
    printf( "%10d%20.12f\n", i, dBernoulli[i] );
  return 0;
}

Output:

n                  Bn
        ----------------------
         0      1.000000000000
         1     -0.500000000000
         2      0.166666666667
         4     -0.033333333333
         6      0.023809523810
         8     -0.033333333333
        10      0.075757575758
        12     -0.253113553114
        14      1.166666666667
        16     -7.092156862745

Parameters

iMax

input maximal index requested

output pointer to the array of numbers declared in the calling module. Array dimension should be iMax+1 or greater