RisingFactorial

The rising factorial $a(a+1)\cdots(a+n-1)$: Fungrim's name for Pochhammer, which it delegates to, extended to a complex non-integer order. Provided by @enumeratio/analytic.

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

RisingFactorial(a, n)the rising factorial $(a)_n = \Gamma(a+n)/\Gamma(a)$.

Typed signatures

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

Domain: Combinatorics

Details

• Evaluates as Pochhammer wherever that head gives an answer, exact or numeric, so the two agree on every shared input. A symbolic order comes back as Pochhammer(a, n).
• Where the native head gives NaN, a complex $a$ with a non-integer order, the value is $\Gamma(a+n)/\Gamma(a)$ through LogGamma. At a pole of $\Gamma(a)$, $a = 0, -1, -2, \dots$, with $n$ off the integers, $1/\Gamma(a) = 0$ makes the value 0, as in Wolfram and mpmath.
• Unlike Pochhammer, it does not thread over a list.

Examples

1 more kept as data, checked against the oracles

Basic #

5 × 6 × 7

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

$(1)_5 = 5!$

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

Scope #

A symbolic base with an integer order expands to the product

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In$\mathrm{RisingFactorial}(a, 3)$Out$a×(a+1)(a+2)$

A rational base: $\tfrac12 \cdot \tfrac32 \cdot \tfrac52$

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

A symbolic order comes back as Pochhammer

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

A complex base and order, through $\Gamma(a+n)/\Gamma(a)$, where native Pochhammer gives NaN

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In$N(\mathrm{RisingFactorial}(2+5i, 3+2i))$Out$20.647\,407\,894\,772\,527+24.047\,356\,710\,232\,72i$

Properties #

The same function as Pochhammer

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In$\mathrm{RisingFactorial}(4, 3)=\mathrm{Pochhammer}(4, 3)$Out$\mathrm{True}$

A negative order gives the reciprocal: $(5)_{-2} = \frac{1}{4 \cdot 3}$

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

Possible issues #

A non-positive integer base with $|a| < n$ picks up a zero factor: $(-3)(-2)(-1)(0)$

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

At a pole of $\Gamma(a)$ with a non-integer order, $1/\Gamma(a) = 0$ makes the value 0

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In$N(\mathrm{RisingFactorial}(-2, 1+i))$Out$0.0$

About

Examples	11
Checked against	Wolfram Language 11 agree
Crosswalk	WikipediaWikidataDLMFFungrimFungrim entry ×22Wolfram Language

Implementation

referenceenginenotatio

The native head, wherever it answers.

nativeenginetypescriptpackages/symbols/analysis/analytic/src/rising-factorial.ts

mappedexternalwolfram / mpmath / sympypackages/oracle/src/mappings.ts

Pochhammer[a, n]; mpmath.rf(a, n); sympy.rf(a, n).

See also: Pochhammer, FallingFactorial, Gamma