\(q\)-Bernoulli Numbers and Polynomials

sage.combinat.q_bernoulli.q_bernoulli(p=None)[source]

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 – 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))               # needs sage.libs.flint
True

>>> from sage.all import *
>>> from sage.combinat.q_bernoulli import q_bernoulli
>>> q_bernoulli(Integer(0))
1
>>> q_bernoulli(Integer(1))
-1/(q + 1)
>>> q_bernoulli(Integer(2))
q/(q^3 + 2*q^2 + 2*q + 1)
>>> all(q_bernoulli(i)(q=Integer(1)) == bernoulli(i) for i in range(Integer(12)))               # needs sage.libs.flint
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)

>>> from sage.all import *
>>> x = PolynomialRing(GF(Integer(2)),'x').gen()
>>> q_bernoulli(Integer(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

>>> from sage.all import *
>>> q_bernoulli(-Integer(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()[source]

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 – 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)           # needs sage.libs.flint
....:     for i in range(12))
True
sage: all(q_bernoulli_polynomial(i)(x=0) == q_bernoulli(i)
....:     for i in range(12))
True

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

>>> from sage.all import *
>>> q_bernoulli_polynomial(-Integer(1))
Traceback (most recent call last):
...
ValueError: the argument must be a nonnegative integer

REFERENCES: [Ca1948], [Ca1954]