bernoulli

Bernoulli Numbers with Odd and Even Indices

The 0th Bernoulli number is 1. The next Bernoulli number can be -1/2 or 1/2, depending on the definition. The bernoulli function uses -1/2. The Bernoulli numbers with even indices n > 1 alternate the signs. Any Bernoulli number with an odd index n > 2 is 0.

Compute the even-indexed Bernoulli numbers with the indices from 0 to 10. Because these indices are not symbolic objects, bernoulli returns floating-point results.

bernoulli(0:2:10)

ans =
    1.0000    0.1667   -0.0333    0.0238   -0.0333    0.0758

Compute the same Bernoulli numbers for the indices converted to symbolic objects:

bernoulli(sym(0:2:10))

ans =
[ 1, 1/6, -1/30, 1/42, -1/30, 5/66]

Compute the odd-indexed Bernoulli numbers with the indices from 1 to 11:

bernoulli(sym(1:2:11))

ans =
[ -1/2, 0, 0, 0, 0, 0]

Bernoulli Polynomials

For the Bernoulli polynomials, use bernoulli with two input arguments.

Compute the first, second, and third Bernoulli polynomials in variables x, y, and z, respectively:

syms x y z
bernoulli(1, x)
bernoulli(2, y)
bernoulli(3, z)

ans =
x - 1/2
 
ans =
y^2 - y + 1/6
 
ans =
z^3 - (3*z^2)/2 + z/2

If the second argument is a number, bernoulli evaluates the polynomial at that number. Here, the result is a floating-point number because the input arguments are not symbolic numbers:

bernoulli(2, 1/3)

ans =
   -0.0556

To get the exact symbolic result, convert at least one of the numbers to a symbolic object:

bernoulli(2, sym(1/3))

ans =
-1/18

Plot Bernoulli Polynomials

Plot the first six Bernoulli polynomials.

syms x
fplot(bernoulli(0:5, x), [-0.8 1.8])
title('Bernoulli Polynomials')
grid on

Handle Expressions Containing Bernoulli Polynomials

Many functions, such as diff and expand, handles expressions containing bernoulli.

Find the first and second derivatives of the Bernoulli polynomial:

syms n x
diff(bernoulli(n,x^2), x)

ans =
2*n*x*bernoulli(n - 1, x^2)

diff(bernoulli(n,x^2), x, x)

ans =
2*n*bernoulli(n - 1, x^2) +...
4*n*x^2*bernoulli(n - 2, x^2)*(n - 1)

Expand these expressions containing the Bernoulli polynomials:

expand(bernoulli(n, x + 3))

ans =
bernoulli(n, x) + (n*(x + 1)^n)/(x + 1) +...
(n*(x + 2)^n)/(x + 2) + (n*x^n)/x

expand(bernoulli(n, 3*x))

ans =
(3^n*bernoulli(n, x))/3 + (3^n*bernoulli(n, x + 1/3))/3 +...
(3^n*bernoulli(n, x + 2/3))/3