\(q\)-Bernoulli Numbers and Polynomials

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

Compute 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
sage.combinat.q_bernoulli.q_bernoulli_polynomial(m)

Compute Carlitz’s \(q\)-analogue of the Bernoulli polynomials.

For every nonnegative integer \(m\), the \(q\)-Bernoulli polynomial is a polynomial in one variable \(x\) with coefficients in \(\QQ(q)\) whose value at \(q=1\) is the usual Bernoulli polynomial \(B_m(x)\).

The original \(q\)-Bernoulli polynomials introduced by Carlitz were polynomials in \(q^y\) with coefficients in \(\QQ(q)\). This function returns these polynomials but expressed in the variable \(x=(q^y-1)/(q-1)\). This allows to let \(q=1\) to recover the classical Bernoulli polynomials.

INPUT:

  • \(m\) – a nonnegative integer

OUTPUT:

A polynomial in one variable \(x\).

EXAMPLES:

sage: from sage.combinat.q_bernoulli import q_bernoulli_polynomial, q_bernoulli
sage: q_bernoulli_polynomial(0)
1
sage: q_bernoulli_polynomial(1)
(2/(q + 1))*x - 1/(q + 1)
sage: x = q_bernoulli_polynomial(1).parent().gen()
sage: all(q_bernoulli_polynomial(i)(q=1)==bernoulli_polynomial(x,i) for i in range(12))
True
sage: all(q_bernoulli_polynomial(i)(x=0)==q_bernoulli(i) for i in range(12))
True

The function does not accept negative arguments:

sage: q_bernoulli_polynomial(-1)
Traceback (most recent call last):
...
ValueError: the argument must be a nonnegative integer

REFERENCES: [Ca1948], [Ca1954]