Pochhammer

The rising factorial (a)_n = a(a+1)(a+2)…(a+n−1).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaFalling and rising factorials↗MathWorldPochhammerSymbol↗WikidataQ2339261↗DLMF5.2↗FungrimRisingFactorial↗Wolfram LanguagePochhammer✓↗

Pochhammer(a, n)the rising factorial $(a)_n$.

SageMathrising_factorial(a, n)SymPyrf(a, n)mpmathrf(a, n)Juliaenumeratio_ball(_rising, a, n)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity) -> number	compute-engine	always
(complex | infinity, complex | infinity) -> number	number-theoryreplaces compute-engine	always

Domain: Combinatorics

Details

• Closed form via the Gamma function: $(a)_n = \frac{\Gamma(a+n)}{\Gamma(a)}$.
• $(1)_n = n!$, since the rising factorial from 1 just counts up to n.
• Negative order gives the falling reciprocal: $(a)_{-n} = \frac{1}{(a-1)(a-2)\cdots(a-n)}$.
• Dividing by $k!$ counts multisets: $\binom{n+k-1}{k} = \frac{(n)_k}{k!}$.
• If a is a non-positive integer with $|a| < n$, the product picks up a zero factor and vanishes.
• A building block of hypergeometric series, where it appears in both numerator and denominator terms.

Plot

ComplexPlot

real slice

Examples

1 more kept as data, checked against the oracles

Basic #

5 × 6 × 7

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In$\mathrm{Pochhammer}(5, 3)$Out$210$

An empty product is 1

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In$\mathrm{Pochhammer}(2, 0)$Out$1$

(1)_5 equals 5!

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

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In$\mathrm{Pochhammer}(3, 4)$Out$360$

10 × 11 × 12 × 13 × 14 × 15

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In$\mathrm{Pochhammer}(10, 6)$Out$3\,603\,600$

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\mathrm{Pochhammer}(\bigl\lbrack2, 3, 4\bigr\rbrack, 2)$Out$\bigl\lbrack6, 12, 20\bigr\rbrack$

A symbolic base with integer order expands to the product

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In$\mathrm{Pochhammer}(n, 5)$Out$n×(n+1)(n+2)(n+3)(n+4)$

A negative order expands to the reciprocal falling product

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In$\mathrm{Pochhammer}(n, -5)$Out$\frac{1}{(n-5)(n-4)(n-3)(n-2)(n-1)}$

Real base and order evaluate as $\frac{\Gamma(a+n)}{\Gamma(a)}$

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In$\mathrm{Pochhammer}(2.4, 8.5)$Out$2\,310\,224.673\,240\,707\,780\,57$

A zero base gives 0 for any positive order

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In$\mathrm{Pochhammer}(0, 1\,285)$Out$0$

A rational base: $\frac12 \cdot \frac32 \cdot \frac52$

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In$\mathrm{Pochhammer}(\frac{1}{2}, 3)$Out$\frac{15}{8}$

A rational order evaluates through Gamma, exactly: $\left(\frac32\right)_{1/2} = \frac{\Gamma(2)}{\Gamma(3/2)} = \frac{2}{\sqrt\pi}$

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In$\mathrm{Pochhammer}(\frac{3}{2}, \frac{1}{2})$Out$\frac{2}{\sqrt{π}}$

Complex base and order evaluate through Gamma

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In$\mathrm{Pochhammer}(2+5i, 8i)$Out$0.000\,002\,138\,682\,291\,868\,096\,3-0.000\,014\,218\,737\,711\,709\,974i$

Dividing by $k!$ counts multisets: $\binom{n+k-1}{k} = \frac{(n)_k}{k!}$, here choosing 3 items from 5 types with repetition

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In$\frac{\mathrm{Pochhammer}(5, 3)}{3!}$Out$35$

$(1)_n = n!$, since the rising factorial from 1 just counts up to n

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In$\mathrm{Pochhammer}(1, 5)=5!$Out$\mathrm{True}$

Negative order gives the reciprocal: $(a)_{-n} = \frac{1}{(a-1)(a-2)\cdots(a-n)}$, so $(5)_{-2} = \frac{1}{4\cdot3}$

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In$\mathrm{Pochhammer}(5, -2)$Out$\frac{1}{12}$

Order 0 is the empty product: $(a)_0 = 1$

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In$\mathrm{Pochhammer}(a, 0)$Out$1$

$(a)_1 = a$

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In$\mathrm{Pochhammer}(a, 1)$Out$a$

The Gamma form $(a)_n = \frac{\Gamma(a+n)}{\Gamma(a)}$. See Gamma

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In$\mathrm{Pochhammer}(4, 3)=\frac{Γ(7)}{Γ(4)}$Out$\mathrm{True}$

Rising and falling factorials: $x^{\underline{n}} = (-1)^n\,(-x)_n$

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In$\mathrm{FallingFactorial}(5, 3)=-\mathrm{Pochhammer}(-5, 3)$Out$\mathrm{True}$

If a is a non-positive integer with $|a| < n$, the product picks up a zero factor: $(-3)(-2)(-1)(0)$

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In$\mathrm{Pochhammer}(-3, 4)$Out$0$

Scope #

$(0)_{-1} = 1/(0-1) = -1$ — exact for any $a \le 0$ and negative integer $n$ (DLMF 5.2.vi)

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In$\mathrm{Pochhammer}(0, -1)$Out$-1$

$(0)_{-2} = 1/[(0-1)(0-2)] = 1/2$

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In$\mathrm{Pochhammer}(0, -2)$Out$\frac{1}{2}$

About

Examples	24
Checked against	Wolfram Language 24 agreeSageMath 22 agree2 errorSymPy 24 agreempmath 20 agreemathlib4 8 agree9 errorJulia 20 agree
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimWolfram 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.ascFactorial n k = n(n+1)...(n+k-1) since Mathlib's 2024 redefinition; natural arguments only, so a rational, real or negative argument fails to typecheck.

See also: Factorial, Binomial