$$q$$-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

$\begin{split}\begin{bmatrix} n \\ k \end{bmatrix}_q = \frac{[n]_q!}{[n-k]_q! \cdot [k]_q!}.\end{split}$

INPUT:

• n, k – the nonnegative integers $$n$$ and $$k$$ defined above

• q – (default: $$q \in \ZZ[q, q^{-1}]$$) 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:

$[n]_q! = [n]_q \cdot [n-1]_q \cdots [2]_q \cdot [1]_q,$

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: $$q \in \ZZ[q, q^{-1}]$$) 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

$[n]_q = \frac{q^n - q^{-n}}{q - q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-n+3} + q^{-n+1}.$

INPUT:

• n – the nonnegative integer $$n$$ defined above

• q – (default: $$q \in \ZZ[q, q^{-1}]$$) 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