$q$-analogs

q_integer(n::IntegerUnion, q::RingElement)
q_integer(n::IntegerUnion)

Return the $q$-integer $[n]_q$ which is defined as $\frac{q^n-1}{q-1}$ when $q-1$ is invertible.

For general ring elements q, we use the following identities to compute $[n]_q$: if n is non-negative, then $[n]_q = \sum_{i=0}^{n-1} q^i$. To handle negative values n we use the identity $[n]_q = -q^{n} [-n]_q$. Thus for negative n we require q to be invertible.

Note that for $q=1$ we obtain $[n]_1 = n$ hence the $q$-integers are "deformations" of the usual integers. For details about these objects see [Con00] or [KC02].

If q is omitted then it defaults to the generator of a Laurent polynomial ring over the integers.

Examples

julia> q_integer(3)
q^2 + q + 1

julia> q_integer(-3)
-q^-1 - q^-2 - q^-3

julia> q_integer(3,2)
7

julia> q_integer(-3,2)
ERROR: DomainError with -3:
Cannot raise an integer x to a negative power -3.

julia> q_integer(-3,2//1)
-7//8

julia> K,i = cyclotomic_field(4, "i");

julia> q_integer(3, i)
i

q_factorial(n::IntegerUnion, q::RingElement)
q_factorial(n::IntegerUnion)

Return the $q$-factorial $[n]_q!$ for a non-negative integer n and an element $q$ of a ring $R$ which is defined as $[1]_q \cdots [n]_q$.

Note that for $q=1$ we obtain $[n]_1! = n!$ hence the $q$-factorial is a "deformation" of the usual factorial. For details about these objects see [Con00] or [KC02].

Examples

julia> q_factorial(3)
q^3 + 2*q^2 + 2*q + 1

julia> q_factorial(3,2)
21

julia> K,i = cyclotomic_field(4, "i");

julia> q_factorial(3, i)
i - 1

q_binomial(n::IntegerUnion, k::IntegerUnion, q::RingElement)
q_binomial(n::IntegerUnion, k::IntegerUnion)

Return the $q$-binomial $\binom{n}{k}_q$ for an integer n and a non-negative integer k which is defined as

\[\binom{n}{k}_q ≔ \frac{[n]_q!}{[k]_q! [n-k]_q!} = \frac{[n]_q [n-1]_q \cdots [n-k+1]_q}{[k]_q!}\]

Note that the first expression is only defined for $n$ greater or equal to $k$ since the $q$-factorials are only defined for non-negative integers, but the second expression is well-defined for all integers n and is used for the implementation.

Note that for $q=1$ we obtain $\binom{n}{k}_1 = \binom{n}{k}$ hence the $q$-binomial coefficient is a "deformation" of the usual binomial coefficient. For details about these objects see [Con00] or [KC02].

Examples

julia> q_binomial(4,2)
q^4 + q^3 + 2*q^2 + q + 1

julia> q_binomial(19,5,-1)
36

julia> K,i = cyclotomic_field(4);

julia> q_binomial(17,10,i)
0

Extended help

In [Con00] it is shown that

\[\binom{n}{k}_q = \sum_{i=0}^{n-k} q^i \binom{i+k-1}{k-1}_q\]

if $n ≥ k > 0$. Since $\binom{n}{0}_q = 1$ for all n and $\binom{n}{k}_q = 0$ if $0 ≤ n < k$ it follows inductively that $\binom{n}{k}_q ∈ ℤ[q]$ if $n ≥ 0$.

For all $n ∈ ℤ$ we have the relation

\[\binom{n}{k}_q = (-1)^k q^{-k(k-1)/2+kn} \binom{k-n-1}{k}_q\]

which shows that $\binom{n}{k}_q ∈ ℤ[q^{-1}]$ if $n < 0$. In particular, $\binom{n}{k}_q ∈ ℤ[q,q^{-1}]$ for all n. Now, for an element $q$ of a ring $R$ we define $\binom{n}{k}_q$ as the specialization of $\binom{n}{k}_q$ in $q$, where $q$ is assumed to be invertible in $R$ if $n < 0$.