-Bernoulli Numbers
Computes Carlitz’s -analogue of the Bernoulli numbers
For every nonnegative integer , the
-Bernoulli number
is a rational function of the indeterminate
whose
value at
is the usual Bernoulli number
.
INPUT:
OUTPUT:
A rational function of the indeterminate (if
is None)
Otherwise, the rational function is evaluated at .
EXAMPLES:
sage: from sage.combinat.q_bernoulli import q_bernoulli
sage: q_bernoulli(0)
1
sage: q_bernoulli(1)
-1/(q + 1)
sage: q_bernoulli(2)
q/(q^3 + 2*q^2 + 2*q + 1)
sage: all(q_bernoulli(i)(q=1)==bernoulli(i) for i in range(12))
True
One can evaluate the rational function by giving a second argument:
sage: x = PolynomialRing(GF(2),'x').gen()
sage: q_bernoulli(5,x)
x/(x^6 + x^5 + x + 1)
The function does not accept negative arguments:
sage: q_bernoulli(-1)
Traceback (most recent call last):
...
ValueError: the argument must be a nonnegative integer.
REFERENCES:
[Ca1948] | Leonard Carlitz, “q-Bernoulli numbers and polynomials”. Duke Math J. 15, 987-1000 (1948), doi:10.1215/S0012-7094-48-01588-9 |
[Ca1954] | Leonard Carlitz, “q-Bernoulli and Eulerian numbers”. Trans Am Soc. 76, 332-350 (1954), doi:10.1090/S0002-9947-1954-0060538-2 |