Euler — Number Theory

Calculates Euler numbers by means of recurrent relation

View versions (2)

Interface

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

using namespace Maths::Discrete::Number_Theory;

void EulerA( int iMax, double* daEuler )
void EulerB( int iMax, double* daEuler )

Overview

Euler numbers, which are also called secant numbers or 'zig' numbers are defined for $|z| <\pi/2$ by

$$\mathrm{sech}(x) - 1 \equiv -\frac{E_1^* x^2}{2!} +\frac{E_2^* x^4}{4!} -\frac{E_3^* x^6}{6!}$$

(1)

The Euler number give the number of odd alternating permutations and are related to Genocchi numbers.

A slightly different convention, which is also used here, defines

$$E_{2n} = (-1)^n E_n^*$$

(2)

$$E_{2n+1} = 0$$

(3)

Thus, the Euler numbers $E_n$ and polynomials $E_n(x)$ are defined through equations

$$\mathrm{sech}(x) = \frac{2{e^z}}{{e^{2z}}+1} = \sum _{n=0}^{\infty}{E_n}\frac{{z^n}}{n!}, |z|< \frac{\pi }{2}$$

(4)

$$\frac{2 e^{xz} }{{e^z}+1} = \sum_{n=0}^{\infty }{E_n}(x)\frac{{z^n}}{n!}, |z| < \pi$$

(5)

The relation between Euler polynomials and numbers:

$${E_n} = {2^n}{E_n}\big(\frac{1}{2}\big)$$

(6)

References

Higher Transcendental Functions, vol.1, (1.14) by H.Bateman and A.Erdelyi (Bateman Manuscript Project), 1953
M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, 1964 chapt.23

FUNCTION

EulerA

Calculates Euler numbers using the recurrent relation

$${E_{2n}} = -\sum_{r=0}^{n-1}\big(\begin{array}{c} 2n \\ 2r \end{array}\big){E_{2r}}, {E_0}=1$$

(7)

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

(8)

Example 1

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

#define MAX_INDEX 16
int main()
{
  double dEulerA[MAX_INDEX+1];
  double dEulerB[MAX_INDEX+1];

  // Table headings
  printf( "%10s %20s %20s\n", "2n", "E_A(2n) (recurrent)", "E_B(2n) (series)" );
  printf( "%8s", " " );
  for( int i = 0; i < 52; i++ )
    printf( "%c", '-' );
  printf( "\n" );

  // array calculation
  Maths::Discrete::Number_Theory::EulerA( MAX_INDEX, dEulerA );
  Maths::Discrete::Number_Theory::EulerB( MAX_INDEX, dEulerB );

  // results output
  for( int i = 0; i <= MAX_INDEX; i+=2 )
    printf( "%10d %20.6f %20.6f\n", i, dEulerA[i], dEulerB[i]);
  return 0;
}

Output:

2n  E_A(2n) (recurrent)     E_B(2n) (series)
----------------------------------------------------
   0             1.000000             1.000000
   2            -1.000000            -1.000000
   4             5.000000             5.000000
   6           -61.000000           -61.000000
   8          1385.000000          1385.000000
  10        -50521.000000        -50521.000000
  12       2702765.000000       2702765.000000
  14    -199360981.000000    -199360981.000000
  16   19391512145.000000   19391512145.000027

Parameters

iMax

input maximal index requested. Must be a power of 2.

daEuler

array into which to place the result, with iMax parameters.

Returns

pointer on the resulting array of type double

Author

Anatoly Prognimack (Mar 19, 2005)

Author

Developed & tested with: Borland C++ 3.1 for DOS and Microsoft Visual C++ 5.0, 6.0

Author

Updated by Will Bateman (March 2005)

FUNCTION

EulerB

Calculates Euler numbers using the recurrent series expansion

$$\begin{array}{l} {E_{2n}}={{(-1)}^n}2(2n)!{{\big(\frac{2}{\pi }\big)}^{2n+1}}\sum _{r=0}^{\infty }\frac{{{(-1)}^r}}{{{(2r+1)}^{2n+1}}}, \\ n=0,1,2,...\\ \end{array}$$

(9)

Parameters

iMax

input maximal index requested. Must be a power of 2.

daEuler

array into which to place the result, with iMax parameters.

Returns

pointer on the resulting array of type double

Author

Anatoly Prognimack (Mar 19, 2005)

Author

Developed & tested with: Borland C++ 3.1 for DOS and Microsoft Visual C++ 5.0, 6.0

Author

Will Bateman (March 2005)