# $$q$$-Bernoulli Numbers¶

$$q$$-Bernoulli Numbers

sage.combinat.q_bernoulli.q_bernoulli(m, p=None)

Computes Carlitz’s $$q$$-analogue of the Bernoulli numbers

For every nonnegative integer $$m$$, the $$q$$-Bernoulli number $$\beta_m$$ is a rational function of the indeterminate $$q$$ whose value at $$q=1$$ is the usual Bernoulli number $$B_m$$.

INPUT:

• $$m$$ – a nonnegative integer
• $$p$$ (default: None) – an optional value for $$q$$

OUTPUT:

A rational function of the indeterminate $$q$$ (if $$p$$ is None)

Otherwise, the rational function is evaluated at $$p$$.

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

q-Analogues

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