FallingFactorial

The falling factorial $x(x-1)\cdots(x-n+1)$ -- Wolfram's own FactorialPower(x, n).

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimFallingFactorial✓↗Fungrim entry1d5e92↗355c22✓↗5b414d✓↗a7b330✓↗e78989✓↗Wolfram LanguageFactorialPower✓↗

FallingFactorial(x, n)exact for an integer x and n, numeric otherwise

SageMathfalling_factorial(x, n)SymPyff(x, n)mpmathff(x, n)Juliaenumeratio_ball((a, n) -> _rising(a - n + 1, n), x, n)

Typed signatures

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

Domain: Wolfram

Plot

Examples

Basic #

$5 \cdot 4 \cdot 3 = 60$

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In$\mathrm{FallingFactorial}(5, 3)$Out$60$

$10\cdot 9 = 90$

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In$\mathrm{FallingFactorial}(10, 2)$Out$90$

A symbolic base expands into a product of 8 factors, written with $-x$

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In$\mathrm{FallingFactorial}(x, 8)$Out$-(x×(1-x)(2-x)(3-x)(4-x)(5-x)(6-x)(7-x))$

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In$\mathrm{FullSimplify}(\operatorname{Series}(\mathrm{FallingFactorial}(x, 5), x, 0, 5))$Out$x\mathrm{FallingFactorial}^{(1, 0)}(0, 5)+\frac{1}{2}\mathrm{FallingFactorial}^{(2, 0)}(0, 5)x^2+\frac{1}{6}\mathrm{FallingFactorial}^{(3, 0)}(0, 5)x^3+\frac{1}{24}\mathrm{FallingFactorial}^{(4, 0)}(0, 5)x^4+\frac{1}{120}\mathrm{FallingFactorial}^{(5, 0)}(0, 5)x^5+O\left(x^6\right)$

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In$\operatorname{Series}(\mathrm{FallingFactorial}(x, 4), x, -1, 3)$Out$24+(x+1)\mathrm{FallingFactorial}^{(1, 0)}(-1, 4)+\frac{1}{2}\mathrm{FallingFactorial}^{(2, 0)}(-1, 4)(x+1)^2+\frac{1}{6}\mathrm{FallingFactorial}^{(3, 0)}(-1, 4)(x+1)^3+O\left((x+1)^4\right)$

Scope #

A factor of $0$ appears once $n$ exceeds $x$

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In$\mathrm{FallingFactorial}(3, 5)$Out$0$

$\tfrac12 \cdot (-\tfrac12) = -\tfrac14$

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In$\mathrm{FallingFactorial}(\frac{1}{2}, 2)$Out$-\frac{1}{4}$

$4\cdot 3 = 12$

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In$\mathrm{FallingFactorial}(4, 2)$Out$12$

FunctionExpand writes the symbolic product out

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In$\mathrm{FunctionExpand}(\mathrm{FallingFactorial}(x, 4))$Out$-(x×(1-x)(2-x)(3-x))$

Zero factors give the empty product 1

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In$\mathrm{FallingFactorial}(n, 0)$Out$1$

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In$\mathrm{FallingFactorial}(x+O\left(x\right)^5, 10)$Out$(-x-O\left(x\right)^5+1)(-x-O\left(x\right)^5+2)(-x-O\left(x\right)^5+3)(-x-O\left(x\right)^5+4)(-x-O\left(x\right)^5+5)(-x-O\left(x\right)^5+6)(-x-O\left(x\right)^5+7)(-x-O\left(x\right)^5+8)(-x-O\left(x\right)^5+9)(-x-O\left(x\right)^5)$

Applications #

$10\cdot 9\cdot 8 = 720$, the number of ordered picks of 3 from 10

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In$\mathrm{FallingFactorial}(10, 3)$Out$720$

Properties #

The forward difference of $x^{\underline n}$ is $n\,x^{\underline{n-1}}$, written as a difference

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In$\mathrm{DifferenceDelta}(\mathrm{FallingFactorial}(x, n), x)$Out$\mathrm{FallingFactorial}(x+1, n)-\mathrm{FallingFactorial}(x, n)$

About

Examples	13
Checked against	Wolfram Language 11 agree2 unevaluatedSageMath 10 agreeSymPy 10 agreempmath 6 agreemathlib4 5 agree1 errorJulia 6 agree
Crosswalk	FungrimFungrim entry ×5Wolfram LanguageSageMathSymPympmathJulia

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

Nat.descFactorial n k = n(n-1)...(n-k+1), which is 0 for k > n like the product through 0; natural arguments only.

See also: RisingFactorial