\(q\)-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)[source]¶
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 aboveq
– (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
>>> from sage.all import * >>> from sage.algebras.quantum_groups.q_numbers import q_binomial >>> q_binomial(Integer(2), Integer(1)) q^-1 + q >>> q_binomial(Integer(2), Integer(0)) 1 >>> q_binomial(Integer(4), Integer(1)) q^-3 + q^-1 + q + q^3 >>> q_binomial(Integer(4), Integer(3)) q^-3 + q^-1 + q + q^3
- sage.algebras.quantum_groups.q_numbers.q_factorial(n, q=None)[source]¶
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 aboveq
– (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
>>> from sage.all import * >>> from sage.algebras.quantum_groups.q_numbers import q_factorial >>> q_factorial(Integer(3)) q^-3 + 2*q^-1 + 2*q + q^3 >>> p = LaurentPolynomialRing(QQ, 'q').gen() >>> q_factorial(Integer(3), p) q^-3 + 2*q^-1 + 2*q + q^3 >>> p = ZZ['p'].gen() >>> q_factorial(Integer(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 (Issue #11411):
sage: q_factorial(-2) Traceback (most recent call last): ... ValueError: argument (-2) must be a nonnegative integer
>>> from sage.all import * >>> q_factorial(-Integer(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)[source]¶
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 aboveq
– (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
>>> from sage.all import * >>> from sage.algebras.quantum_groups.q_numbers import q_int >>> q_int(Integer(2)) q^-1 + q >>> q_int(Integer(3)) q^-2 + 1 + q^2 >>> q_int(Integer(5)) q^-4 + q^-2 + 1 + q^2 + q^4 >>> q_int(Integer(5), Integer(1)) 5