QFactorial

The q-factorial $[n]_q! = [1]_q [2]_q \cdots [n]_q$, where $[k]_q = 1 + q + \cdots + q^{k-1}$. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageQFactorial✓↗

QFactorial(n, q)the q-factorial $[n]_q!$.

SageMathenumeratio_q_factorial(n, q)

Typed signatures

Type	Package	Applies
(real, complex) -> number	analytic	always

Domain: Special functions

Details

• Built as a product of q-integers $[k]_q$, each a sum of $k$ powers of $q$ — exact boxed arithmetic throughout, so it reduces to a number for numeric $q$ (rational stays rational) and to a genuine polynomial in $q$ that Expand can open up for symbolic $q$.
• At $q = 1$, $[k]_1 = k$ termwise (no division, so no $q \to 1$ limit to take), reducing exactly to $n!$.
• A non-integer $n$ is the q-Gamma function, $[n]_q! = \Gamma_q(n+1)$, which N() evaluates for real $q > 0$ through the infinite products $(q;q)_\infty / \big((q^{n+1};q)_\infty (1-q)^n\big)$ (for $q > 1$ reflected by $[n]_q! = q^{n(n-1)/2}[n]_{1/q}!$, and $\Gamma(n+1)$ at $q = 1$). A negative integer $n$ is a pole, and $q \le 0$ stays symbolic.
• A symbolic, non-integer or negative $n$ is a held call (still canonical). At $q = 0$ with a non-integer rational $n$, Series expands the same product quotient as a series in $q^{1/d}$, with Wolfram's truncation; for $-2 < n < -1$ the leading factor makes it start at a positive fractional power, and a lower $n$ stays unevaluated.

Plot

the complex plot cannot lower QFactorial

Examples

4 more kept as data, checked against the oracles

Basic #

$[1]_2 [2]_2 [3]_2 = 1 \cdot 3 \cdot 7$

live ↗

In$\mathrm{QFactorial}(3, 2)$Out$21$

live ↗

In$\mathrm{QFactorial}(4, 2)$Out$315$

live ↗

In$\operatorname{Series}((q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1)(q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1)(q^7+q^6+q^5+q^4+q^3+q^2+q+1)(q^6+q^5+q^4+q^3+q^2+q+1)(q^5+q^4+q^3+q^2+q+1)(q^4+q^3+q^2+q+1)(q^3+q^2+q+1)(q^2+q+1)(q+1), q, 0, 10)$Out$1+9q+44q^2+155q^3+440q^4+1\,068q^5+2\,298q^6+4\,489q^7+8\,095q^8+13\,640q^9+21\,670q^{10}+O\left(q^{11}\right)$

live ↗

In$\operatorname{Series}((x^2+x+1)(x+1), x, ∞, 6)$Out$x^3+2x^2+2x+1$

Scope #

live ↗

In$\mathrm{QFactorial}(3, \frac{1}{2})$Out$\frac{21}{8}$

A matrix function of a polynomial gives 0 at $-1$ and 186 at 5

live ↗

In$\mathrm{MatrixFunction}(\operatorname{\_1}↦(\operatorname{\_1}^2+\operatorname{\_1}+1)(\operatorname{\_1}+1), \bigl\lbrack\bigl\lbrack-1, 0\bigr\rbrack, \bigl\lbrack0, 5\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack0, 0\bigr\rbrack, \bigl\lbrack0, 186\bigr\rbrack\bigr\rbrack$

live ↗

In$N(\mathrm{QFactorial}(\frac{1}{3}, 6), 50)$Out$0.786\,862\,823\,421\,685\,819\,914\,448\,763\,102\,170\,857\,654\,681\,413\,752\,97$

live ↗

In$\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{QFactorial}(x, q), x, 0, 3)))$Out$\frac{1}{6}\mathrm{QFactorial}^{(3, 0)}(0, q)x^3+\frac{1}{2}\mathrm{QFactorial}^{(2, 0)}(0, q)x^2+x\mathrm{QFactorial}^{(1, 0)}(0, q)+1$

live ↗

In$\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{QFactorial}(x, q), x, \mathrm{x0}, 2)))$Out$\frac{1}{2}\frac{∂^{2}}{∂\mathrm{x0}^{2}} \mathrm{QFactorial}(\mathrm{x0}, q)(x-\mathrm{x0})^2+(x-\mathrm{x0})\frac{∂}{∂\mathrm{x0}} \mathrm{QFactorial}(\mathrm{x0}, q)+\mathrm{QFactorial}(\mathrm{x0}, q)$

Applications #

The inversion generating function of SymmetricGroup(3): $\sum_\pi q^{\mathrm{inv}(\pi)}$

live ↗

In$\mathrm{Expand}(\mathrm{QFactorial}(3, q))$Out$q^3+2q^2+2q+1$

Properties #

At $q = 1$ it is the ordinary factorial

live ↗

In$\mathrm{QFactorial}(5, 1)$Out$120$

A non-integer argument gives a series in $q^{1/2}$

live ↗

In$\operatorname{Series}(\mathrm{QFactorial}(\frac{1}{2}, q), q, 0, 3)$Out$-\frac{1}{8}×5\sqrt{q}^{7}+\frac{\sqrt{q}^{5}}{2}+\sqrt{q}^{3}+1-\frac{q}{2}-\frac{9q^2}{8}+\frac{7q^3}{16}+O\left(q^4\right)$

A negative order starts the series at a positive fractional power

live ↗

In$\operatorname{Series}(\mathrm{QFactorial}(-\frac{3}{2}, q), q, 0, 3)$Out$-\frac{\sqrt{q}^{3}}{2}-\frac{1}{8}×3\sqrt{q}^{5}-\sqrt{q}-2q+\frac{3q^3}{4}+O\left(\sqrt{q}^{7}\right)$

About

Examples	17
Checked against	Wolfram Language 17 agreeSageMath 9 agree2 unevaluated5 errorSymPy 2 agree1 error
Crosswalk	Wolfram LanguageSageMath

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/q-series.ts

product of q-integers, built at canonicalization time (not evaluate) so Expand sees the tree to open up.

mappedexternalwolframpackages/oracle/src/mappings.ts

QFactorial[n, q].

mappedsagepackages/oracle/src/mappings.ts

See also: QPochhammer, QBinomial, Factorial