$q$ -Numbers#

Note

These are the quantum group $q$ -analogs, not the usual $q$ -analogs typically used in combinatorics (see sage.combinat.q_analogues).

sage.algebras.quantum_groups.q_numbers.q_binomial(n, k, q=None)#

Return the $q$ -binomial coefficient.

Let $[n]_{q}!$ denote the $q$ -factorial of $n$ given by sage.algebras.quantum_groups.q_numbers.q_factorial(). The $q$ -binomial coefficient is defined by

\begin{array}{r} {[\begin{array}{c} n \\ k \end{array}]}_{q} = \frac{[n]_{q}!}{[n - k]_{q}! \cdot [k]_{q}!} . \end{array}

INPUT:

n, k – the nonnegative integers $n$ and $k$ defined above
q – (default: $q \in Z [q, q^{- 1}]$ ) the parameter $q$ (should be invertible)

If q is unspecified, then it is taken to be the generator $q$ for a Laurent polynomial ring over the integers.

Note

This is not the “usual” $q$ -binomial but a variant useful for quantum groups. For the version used in combinatorics, see sage.combinat.q_analogues.

Warning

This method uses division by $q$ -factorials. If $[k]_{q}!$ or $[n - k]_{q}!$ are zero-divisors, or division is not implemented in the ring containing $q$ , then it will not work.

EXAMPLES:

sage: from sage.algebras.quantum_groups.q_numbers import q_binomial
sage: q_binomial(2, 1)
q^-1 + q
sage: q_binomial(2, 0)
1
sage: q_binomial(4, 1)
q^-3 + q^-1 + q + q^3
sage: q_binomial(4, 3)
q^-3 + q^-1 + q + q^3

sage.algebras.quantum_groups.q_numbers.q_factorial(n, q=None)#

Return the $q$ -analog of the factorial $n!$ .

The $q$ -factorial is defined by:

[n]_{q}! = [n]_{q} \cdot [n - 1]_{q} \dots [2]_{q} \cdot [1]_{q},

where $[n]_{q}$ denotes the $q$ -integer defined in sage.algebras.quantum_groups.q_numbers.q_int().

INPUT:

n – the nonnegative integer $n$ defined above
q – (default: $q \in Z [q, q^{- 1}]$ ) the parameter $q$ (should be invertible)

If q is unspecified, then it defaults to using the generator $q$ for a Laurent polynomial ring over the integers.

Note

This is not the “usual” $q$ -factorial but a variant useful for quantum groups. For the version used in combinatorics, see sage.combinat.q_analogues.

EXAMPLES:

sage: from sage.algebras.quantum_groups.q_numbers import q_factorial
sage: q_factorial(3)
q^-3 + 2*q^-1 + 2*q + q^3
sage: p = LaurentPolynomialRing(QQ, 'q').gen()
sage: q_factorial(3, p)
q^-3 + 2*q^-1 + 2*q + q^3
sage: p = ZZ['p'].gen()
sage: q_factorial(3, p)
(p^6 + 2*p^4 + 2*p^2 + 1)/p^3

The $q$ -analog of $n!$ is only defined for $n$ a nonnegative integer (trac ticket #11411):

sage: q_factorial(-2)
Traceback (most recent call last):
...
ValueError: argument (-2) must be a nonnegative integer

sage.algebras.quantum_groups.q_numbers.q_int(n, q=None)#

Return the $q$ -analog of the nonnegative integer $n$ .

The $q$ -analog of the nonnegative integer $n$ is given by

[n]_{q} = \frac{q^{n} - q^{- n}}{q - q^{- 1}} = q^{n - 1} + q^{n - 3} + \dots + q^{- n + 3} + q^{- n + 1} .

INPUT:

n – the nonnegative integer $n$ defined above
q – (default: $q \in Z [q, q^{- 1}]$ ) the parameter $q$ (should be invertible)

If q is unspecified, then it defaults to using the generator $q$ for a Laurent polynomial ring over the integers.

Note

This is not the “usual” $q$ -analog of $n$ (or $q$ -integer) but a variant useful for quantum groups. For the version used in combinatorics, see sage.combinat.q_analogues.

EXAMPLES:

sage: from sage.algebras.quantum_groups.q_numbers import q_int
sage: q_int(2)
q^-1 + q
sage: q_int(3)
q^-2 + 1 + q^2
sage: q_int(5)
q^-4 + q^-2 + 1 + q^2 + q^4
sage: q_int(5, 1)
5

q-Numbers#

$q$ -Numbers#